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Annotation of OpenXM/src/asir-contrib/packages/doc/sm1.oxweave, Revision 1.11

1.11    ! takayama    1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.10 2003/05/20 23:25:28 takayama Exp $ */
1.1       takayama    2:
                      3: /*&C-texi
                      4: @c DO NOT EDIT THIS FILE   oxphc.texi
                      5: */
1.6       takayama    6: /*&C-texi
                      7: @node SM1 Functions,,, Top
1.11    ! takayama    8:
1.6       takayama    9: */
1.1       takayama   10: /*&jp-texi
                     11: @chapter SM1 $BH!?t(B
                     12:
                     13: $B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}
                     14: $B$H$N%$%s%?%U%'!<%94X?t$r2r@b$9$k(B.
                     15: $B$3$l$i$N4X?t$O%U%!%$%k(B  @file{sm1} $B$GDj5A$5$l$F$$$k(B.
                     16: @file{sm1} $B$O(B @file{$(OpenXM_HOME)/lib/asir-contrib} $B$K$"$k(B.
                     17: $B%7%9%F%`(B @code{sm1} $B$OHyJ,:nMQAG4D$G7W;;$9$k$?$a$N%7%9%F%`$G$"$k(B.
                     18: $B7W;;Be?t4v2?$N$$$m$$$m$JITJQNL$N7W;;$,HyJ,:nMQAG$N7W;;$K5"Ce$9$k(B.
                     19: @code{sm1} $B$K$D$$$F$NJ8=q$O(B @code{OpenXM/doc/kan96xx} $B$K$"$k(B.
                     20:
                     21: $B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,
                     22: $BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.
                     23: $B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.
                     24:
                     25: @tex
                     26: $B6u4V(B
                     27: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$
                     28: $B$N%I%i!<%`%3%[%b%m%872C#$N<!85$r7W;;$7$F$_$h$&(B.
                     29: $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE@(B $x=0$, $x=1$ $B$N$^$o$j$r(B
                     30: $B$^$o$kFs$D$N%k!<%W$,(B1$B<!85$N%[%b%m%8!<72$N6u4V$r$O$k(B.
                     31: $B$7$?$,$C$F(B, 1$B<!85%I%i!<%`%3%[%b%m%872$N<!85$O(B $2$ $B$G$"$k(B.
                     32: @code{sm1} $B$O(B $0$ $B<!85$N%3%[%b%m%872$N<!85$*$h$S(B $1$ $B<!85$N%3%[%b%m%872$N(B
                     33: $B<!85$rEz$($k(B.
                     34: @end tex
                     35: */
                     36: /*&eg-texi
                     37: @chapter SM1 Functions
                     38:
                     39: This chapter describes  interface functions for
                     40: sm1 ox server @code{ox_sm1_forAsir}.
                     41: These interface functions are defined in the file @file{sm1}.
                     42: The file @file{sm1} is @*
                     43: at @file{$(OpenXM_HOME)/lib/asir/contrib-packages}.
                     44: The system @code{sm1} is a system to compute in the ring of differential
                     45: operators.
                     46: Many constructions of invariants
                     47: in the computational algebraic geometry reduce
                     48: to constructions in the ring of differential operators.
                     49: Documents on @code{sm1} are in
                     50: the directory @code{OpenXM/doc/kan96xx}.
                     51:
                     52: All the coefficients of input polynomials should be
                     53: integers for most functions in this section.
                     54: Other functions accept rational numbers as inputs
                     55: and it will be explicitely noted in each explanation
                     56: of these functions.
                     57:
                     58:
                     59:
                     60: @tex
                     61: Let us evaluate the dimensions of the de Rham cohomology groups
                     62: of
                     63: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$.
                     64: The space $X$ is a two punctured plane, so two loops that encircles the
                     65: points $x=0$ and $x=1$ respectively spans the first homology group.
                     66: Hence, the dimension of the first de Rham cohomology group is $2$.
                     67: @code{sm1} answers the dimensions of the 0th and the first
                     68: cohomology groups.
                     69: @end tex
                     70: */
                     71: /*&C-texi
                     72: @example
                     73:
1.5       takayama   74: @include opening.texi
1.1       takayama   75:
1.8       takayama   76: [283] sm1.deRham([x*(x-1),[x]]);
1.1       takayama   77: [1,2]
                     78: @end example
                     79: */
                     80: /*&C-texi
                     81: @noindent
                     82: The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*
                     83: The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
                     84: Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
                     85: Grobner Deformations of Hypergeometric Differential Equations,
                     86: 1999, Springer.
                     87: See the appendix.
                     88: */
1.6       takayama   89:
1.11    ! takayama   90: /*&C-texi
1.6       takayama   91: @menu
                     92: * ox_sm1_forAsir::
1.8       takayama   93: * sm1.start::
1.10      takayama   94: * sm1.sm1::
1.8       takayama   95: * sm1.push_int0::
                     96: * sm1.gb::
                     97: * sm1.deRham::
                     98: * sm1.hilbert::
                     99: * sm1.genericAnn::
                    100: * sm1.wTensor0::
                    101: * sm1.reduction::
                    102: * sm1.xml_tree_to_prefix_string::
                    103: * sm1.syz::
                    104: * sm1.mul::
                    105: * sm1.distraction::
                    106: * sm1.gkz::
                    107: * sm1.appell1::
                    108: * sm1.appell4::
                    109: * sm1.rank::
                    110: * sm1.auto_reduce::
                    111: * sm1.slope::
1.11    ! takayama  112: * sm1.gb_d::
        !           113: * sm1.syz_d::
        !           114: * sm1.ahg::
        !           115: * sm1.bfunction::
        !           116: * sm1.generalized_bfunction::
        !           117: * sm1.restriction::
        !           118: * sm1.saturation::
1.6       takayama  119: @end menu
                    120: */
                    121:
1.1       takayama  122: /*&jp-texi
                    123: @section @code{ox_sm1_forAsir} $B%5!<%P(B
                    124: */
                    125: /*&eg-texi
                    126: @section @code{ox_sm1_forAsir} Server
                    127: */
                    128:
                    129: /*&eg-texi
1.11    ! takayama  130: @node ox_sm1_forAsir,,, SM1 Functions
1.1       takayama  131: @subsection @code{ox_sm1_forAsir}
                    132: @findex ox_sm1_forAsir
                    133: @table @t
                    134: @item ox_sm1_forAsir
                    135: ::  @code{sm1} server for @code{asir}.
                    136: @end table
                    137: @itemize @bullet
                    138: @item
                    139:    @code{ox_sm1_forAsir} is the @code{sm1} server started from asir
1.8       takayama  140:     by the command @code{sm1.start}.
1.1       takayama  141:     In the standard setting,  @*
                    142:     @code{ox_sm1_forAsir} =
                    143:          @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
                    144:        +
                    145:          @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1}   (macro file) @*
                    146:        +
                    147:          @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1}  (macro file) @*
                    148:    The macro files @file{callsm1.sm1} and @file{callsm1b.sm1}
                    149:    are searched from
                    150:    current directory, @code{$(LOAD_SM1_PATH)},
                    151:    @code{$(OpenXM_HOME)/lib/sm1},
                    152:    @code{/usr/local/lib/sm1}
                    153:    in this order.
                    154: @item Note for programmers:  See the files
                    155:     @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
                    156:     @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
                    157: to build your own server by reading @code{sm1} macros.
                    158: @end itemize
                    159: */
                    160: /*&jp-texi
1.11    ! takayama  161: @node ox_sm1_forAsir,,, SM1 Functions
1.1       takayama  162: @subsection @code{ox_sm1_forAsir}
                    163: @findex ox_sm1_forAsir
                    164: @table @t
                    165: @item ox_sm1_forAsir
                    166: ::  @code{asir} $B$N$?$a$N(B @code{sm1} $B%5!<%P(B.
                    167: @end table
                    168: @itemize @bullet
                    169: @item
                    170:    $B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B
1.8       takayama  171:     @code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.
1.1       takayama  172:
                    173:     $BI8=`E*@_Dj$G$O(B, @*
                    174:     @code{ox_sm1_forAsir} =
                    175:          @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
                    176:        +
                    177:          @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1}   (macro file) @*
                    178:        +
                    179:          @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1}  (macro file) @*
                    180:   $B$G$"$j(B, $B$3$l$i$N%^%/%m%U%!%$%k$O(B, $B0lHL$K$O(B
                    181:    current directory, @code{$(LOAD_SM1_PATH)},
                    182:    @code{$(OpenXM_HOME)/lib/sm1},
                    183:    @code{/usr/local/lib/sm1}
                    184:     $B$N=gHV$G$5$,$5$l$k(B.
                    185: @item $B%W%m%0%i%^!<$N$?$a$N%N!<%H(B:
                    186: @code{sm1} $B%^%/%m$rFI$_9~$s$G<+J,FH<+$N%5!<%P$r:n$k$K$O(B
                    187:     $B<!$N%U%!%$%k$b8+$h(B
                    188:     @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
                    189:     @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
                    190: @end itemize
                    191: */
                    192:
                    193:
                    194: /*&jp-texi
                    195: @section $BH!?t0lMw(B
                    196: */
                    197: /*&eg-texi
                    198: @section Functions
                    199: */
                    200:
                    201: /*&eg-texi
1.8       takayama  202: @c sort-sm1.start
                    203: @node sm1.start,,, SM1 Functions
                    204: @subsection @code{sm1.start}
                    205: @findex sm1.start
1.1       takayama  206: @table @t
1.8       takayama  207: @item sm1.start()
1.1       takayama  208: ::  Start  @code{ox_sm1_forAsir} on the localhost.
                    209: @end table
                    210:
                    211: @table @var
                    212: @item return
                    213: Integer
                    214: @end table
                    215:
                    216: @itemize @bullet
                    217: @item Start @code{ox_sm1_forAsir} on the localhost.
                    218:     It returns the descriptor of @code{ox_sm1_forAsir}.
                    219: @item Set @code{Xm_noX = 1} to start @code{ox_sm1_forAsir}
                    220: without a debug window.
                    221: @item You might have to set suitable orders of variable by the command
                    222: @code{ord}.  For example,
                    223: when you are working in the
                    224: ring of differential operators on the variable @code{x} and @code{dx}
                    225: (@code{dx} stands for
                    226: @tex $\partial/\partial x$
                    227: @end tex
                    228: ),
                    229: @code{sm1} server assumes that
                    230: the variable @code{dx} is collected to the right and the variable
                    231: @code{x} is collected to the left in the printed expression.
                    232: In the example below, you  must not use the variable @code{cc}
                    233: for computation in @code{sm1}.
                    234: @item The variables from @code{a} to @code{z} except @code{d} and @code{o}
                    235: and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
                    236: @code{z0}, ..., @code{z20} can be used as variables for ring of
                    237: differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).
1.8       takayama  238: @item The descriptor is stored in @code{static Sm1_proc}.
                    239: The descriptor can be obtained by the function
                    240: @code{sm1.get_Sm1_proc()}.
1.1       takayama  241: @end itemize
                    242: */
                    243: /*&jp-texi
1.8       takayama  244: @c sort-sm1.start
                    245: @node sm1.start,,, SM1 Functions
                    246: @subsection @code{sm1.start}
                    247: @findex sm1.start
1.1       takayama  248: @table @t
1.8       takayama  249: @item sm1.start()
1.1       takayama  250: ::  localhost $B$G(B  @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
                    251: @end table
                    252:
                    253: @table @var
                    254: @item return
                    255: $B@0?t(B
                    256: @end table
                    257:
                    258: @itemize @bullet
                    259: @item localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
                    260: $B%5!<%P(B @code{ox_sm1_forAsir} $B$N<1JLHV9f$rLa$9(B.
                    261: @item @code{Xm_noX = 1} $B$H$*$/$H%5!<%P(B @code{ox_sm1_forAsir} $B$r%G%P%C%0MQ$N(B
                    262: $B%&%#%s%I%&$J$7$K5/F0$G$-$k(B.
                    263: @item $B%3%^%s%I(B @code{ord} $B$rMQ$$$FJQ?t=g=x$r@5$7$/@_Dj$7$F$*$/I,MW$,(B
                    264: $B$"$k(B.
                    265: $B$?$H$($P(B,
                    266: $BJQ?t(B @code{x} $B$H(B @code{dx} $B>e$NHyJ,:nMQAG4D(B
                    267: (@code{dx} $B$O(B
                    268: @tex $\partial/\partial x$
                    269: @end tex
                    270: $B$KBP1~(B)
                    271: $B$G7W;;$7$F$$$k$H$-(B,
                    272: @code{sm1} $B%5!<%P$O<0$r0u:~$7$?$H$-(B,
                    273: $BJQ?t(B @code{dx} $B$O1&B&$K=8$a$lJQ?t(B
                    274: @code{x} $B$O:8B&$K$"$D$a$i$l$F$$$k$H2>Dj$7$F$$$k(B.
                    275: $B<!$NNc$G$O(B, $BJQ?t(B @code{cc} $B$r(B @code{sm1} $B$G$N7W;;$N$?$a$KMQ$$$F$O$$$1$J$$(B.
                    276: @item @code{a} $B$h$j(B @code{z} $B$N$J$+$G(B,  @code{d} $B$H(B @code{o} $B$r=|$$$?$b$N(B,
                    277: $B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
                    278: @code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B
                    279: $B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).
1.8       takayama  280: @item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.
                    281: $B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.
1.1       takayama  282: @end itemize
                    283: */
                    284: /*&C-texi
                    285: @example
                    286: [260] ord([da,a,db,b]);
                    287: [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
                    288: ......... omit ..................
                    289: ]
                    290: [261] a*da;
                    291: a*da
                    292: [262] cc*dcc;
                    293: dcc*cc
1.8       takayama  294: [263] sm1.mul(da,a,[a]);
1.1       takayama  295: a*da+1
1.8       takayama  296: [264] sm1.mul(a,da,[a]);
1.1       takayama  297: a*da
                    298: @end example
                    299: */
                    300: /*&eg-texi
                    301: @table @t
                    302: @item Reference
1.8       takayama  303:     @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
1.1       takayama  304:     @code{ord}
                    305: @end table
                    306: */
                    307: /*&jp-texi
                    308: @table @t
                    309: @item $B;2>H(B
1.8       takayama  310:     @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
1.1       takayama  311:     @code{ord}
                    312: @end table
                    313: */
                    314:
                    315:
                    316:
                    317: /*&eg-texi
                    318: @c sort-sm1
1.10      takayama  319: @node sm1.sm1,,, SM1 Functions
                    320: @subsection @code{sm1.sm1}
                    321: @findex sm1.sm1
1.1       takayama  322: @table @t
1.10      takayama  323: @item sm1.sm1(@var{p},@var{s})
1.1       takayama  324: ::  ask the @code{sm1} server to execute the command string @var{s}.
                    325: @end table
                    326:
                    327: @table @var
                    328: @item return
                    329: Void
                    330: @item p
                    331: Number
                    332: @item s
                    333: String
                    334: @end table
                    335:
                    336: @itemize @bullet
                    337: @item  It asks the @code{sm1} server of the descriptor number @var{p}
                    338: to execute the command string @var{s}.
1.9       takayama  339: (In the next example, the descriptor number is 0.)
1.1       takayama  340: @end itemize
                    341: */
                    342: /*&jp-texi
1.10      takayama  343: @node sm1.sm1,,, SM1 Functions
                    344: @subsection @code{sm1.sm1}
                    345: @findex sm1.sm1
1.1       takayama  346: @table @t
1.10      takayama  347: @item sm1.sm1(@var{p},@var{s})
1.1       takayama  348: ::  $B%5!<%P(B @code{sm1} $B$K%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$K$?$N$`(B.
                    349: @end table
                    350:
                    351: @table @var
                    352: @item return
                    353: $B$J$7(B
                    354: @item p
                    355: $B?t(B
                    356: @item s
                    357: $BJ8;zNs(B
                    358: @end table
                    359:
                    360: @itemize @bullet
                    361: @item  $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B
                    362: $B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.
1.9       takayama  363:  ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
1.1       takayama  364: @end itemize
                    365: */
                    366: /*&C-texi
                    367: @example
1.10      takayama  368: [261] sm1.sm1(0," ( (x-1)^2 ) . ");
1.1       takayama  369: 0
                    370: [262] ox_pop_string(0);
                    371: x^2-2*x+1
1.10      takayama  372: [263] sm1.sm1(0," [(x*(x-1))  [(x)]] deRham ");
1.1       takayama  373: 0
                    374: [264] ox_pop_string(0);
                    375: [1 , 2]
                    376: @end example
                    377: */
1.10      takayama  378:
1.1       takayama  379: /*&jp-texi
                    380: @table @t
                    381: @item $B;2>H(B
1.8       takayama  382:     @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
1.1       takayama  383: @end table
                    384: */
                    385: /*&eg-texi
                    386: @table @t
                    387: @item Reference
1.8       takayama  388:     @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
1.1       takayama  389: @end table
                    390: */
                    391:
                    392:
                    393: /*&eg-texi
1.8       takayama  394: @c sort-sm1.push_int0
                    395: @node sm1.push_int0,,, SM1 Functions
                    396: @subsection @code{sm1.push_int0}
                    397: @findex sm1.push_int0
1.1       takayama  398: @table @t
1.8       takayama  399: @item sm1.push_int0(@var{p},@var{f})
1.1       takayama  400: ::   push the object @var{f} to the server with the descriptor number @var{p}.
                    401: @end table
                    402:
                    403: @table @var
                    404: @item return
                    405: Void
                    406: @item p
                    407: Number
                    408: @item f
                    409: Object
                    410: @end table
                    411:
                    412: @itemize @bullet
                    413: @item When @code{type(@var{f})} is 2 (recursive polynomial),
                    414:     @var{f} is converted to a string (type == 7)
                    415:     and is sent to the server by @code{ox_push_cmo}.
                    416: @item When @code{type(@var{f})} is 0 (zero),
                    417:      it is translated to the 32 bit integer zero
                    418:     on the server.
                    419:     Note that @code{ox_push_cmo(@var{p},0)} sends @code{CMO_NULL} to the server.
                    420: In other words, the server does not get the 32 bit integer 0 nor
                    421: the bignum 0.
                    422: @item  @code{sm1} integers are classfied into the 32 bit integer and
                    423: the bignum.
                    424: When @code{type(@var{f})} is 1 (number), it is translated to the
                    425: 32 bit integer on the server.
                    426: Note that @code{ox_push_cmo(@var{p},1234)} send the bignum 1234 to the
                    427: @code{sm1} server.
                    428: @item In other cases,  @code{ox_push_cmo} is called without data conversion.
                    429: @end itemize
                    430: */
                    431: /*&jp-texi
1.8       takayama  432: @c sort-sm1.push_int0
                    433: @node sm1.push_int0,,, SM1 Functions
                    434: @subsection @code{sm1.push_int0}
                    435: @findex sm1.push_int0
1.1       takayama  436: @table @t
1.8       takayama  437: @item sm1.push_int0(@var{p},@var{f})
1.1       takayama  438: ::   $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.
                    439: @end table
                    440:
                    441: @table @var
                    442: @item return
                    443: $B$J$7(B
                    444: @item p
                    445: $B?t(B
                    446: @item f
                    447: $B%*%V%8%'%/%H(B
                    448: @end table
                    449:
                    450: @itemize @bullet
                    451: @item @code{type(@var{f})} $B$,(B 2 ($B:F5"B?9`<0(B) $B$N$H$-(B,
                    452:     @var{f} $B$OJ8;zNs(B (type == 7) $B$KJQ49$5$l$F(B,
                    453:     @code{ox_push_cmo} $B$rMQ$$$F%5!<%P$XAw$i$l$k(B.
                    454: @item @code{type(@var{f})} $B$,(B 0 (zero) $B$N$H$-$O(B,
                    455:     $B%5!<%P>e$G$O(B, 32 bit $B@0?t$H2r<a$5$l$k(B.
                    456:     $B$J$*(B @code{ox_push_cmo(P,0)} $B$O%5!<%P$KBP$7$F(B @code{CMO_NULL}
                    457: $B$r$*$/$k$N$G(B, $B%5!<%PB&$G$O(B, 32 bit $B@0?t$r<u$1<h$k$o$1$G$O$J$$(B.
                    458: @item  @code{sm1} $B$N@0?t$O(B, 32 bit $B@0?t$H(B bignum $B$K$o$1$k$3$H$,$G$-$k(B.
                    459: @code{type(@var{f})} $B$,(B 1 ($B?t(B)$B$N$H$-(B, $B$3$N4X?t$O(B 32 bit integer $B$r%5!<%P$K(B
                    460: $B$*$/$k(B.
                    461: @code{ox_push_cmo(@var{p},1234)} $B$O(B bignum $B$N(B 1234 $B$r(B
                    462: @code{sm1} $B%5!<%P$K$*$/$k$3$H$KCm0U$7$h$&(B.
                    463: @item $B$=$NB>$N>l9g$K$O(B  @code{ox_push_cmo} $B$r%G!<%?7?$NJQ49$J$7$K8F$S=P$9(B.
                    464: @end itemize
                    465: */
                    466: /*&C
                    467: @example
1.8       takayama  468: [219] P=sm1.start();
1.1       takayama  469: 0
1.8       takayama  470: [220] sm1.push_int0(P,x*dx+1);
1.1       takayama  471: 0
                    472: [221] A=ox_pop_cmo(P);
                    473: x*dx+1
                    474: [223] type(A);
                    475: 7   (string)
                    476: @end example
                    477:
                    478: @example
1.8       takayama  479: [271] sm1.push_int0(0,[x*(x-1),[x]]);
1.1       takayama  480: 0
                    481: [272] ox_execute_string(0," deRham ");
                    482: 0
                    483: [273] ox_pop_cmo(0);
                    484: [1,2]
                    485: @end example
                    486: */
                    487: /*&eg-texi
                    488: @table @t
                    489: @item Reference
                    490:     @code{ox_push_cmo}
                    491: @end table
                    492: */
                    493: /*&jp-texi
                    494: @table @t
                    495: @item Reference
                    496:     @code{ox_push_cmo}
                    497: @end table
                    498: */
                    499:
                    500:
                    501:
                    502: /*&eg-texi
1.8       takayama  503: @c sort-sm1.gb
                    504: @node sm1.gb,,, SM1 Functions
                    505: @node sm1.gb_d,,, SM1 Functions
                    506: @subsection @code{sm1.gb}
                    507: @findex sm1.gb
                    508: @findex sm1.gb_d
1.1       takayama  509: @table @t
1.8       takayama  510: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
1.1       takayama  511: ::  computes the Grobner basis of @var{f} in the ring of differential
                    512: operators with the variable @var{v}.
1.8       takayama  513: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1       takayama  514: ::  computes the Grobner basis of @var{f} in the ring of differential
                    515: operators with the variable @var{v}.
                    516: The result will be returned as a list of distributed polynomials.
                    517: @end table
                    518:
                    519: @table @var
                    520: @item return
                    521: List
1.3       takayama  522: @item p, q, r
1.1       takayama  523: Number
                    524: @item f, v, w
                    525: List
                    526: @end table
                    527:
                    528: @itemize @bullet
                    529: @item
                    530:    It returns the Grobner basis of the set of polynomials @var{f}
                    531:    in the ring of deferential operators with the variables @var{v}.
                    532: @item
                    533:    The weight vectors are given by @var{w}, which can be omitted.
                    534:     If @var{w} is not given,
                    535:     the graded reverse lexicographic order will be used to compute Grobner basis.
                    536: @item
1.8       takayama  537:    The return value of @code{sm1.gb}
1.1       takayama  538:     is the list of the Grobner basis of @var{f} and the initial
                    539:     terms (when @var{w} is not given) or initial ideal (when @var{w} is given).
                    540: @item
1.8       takayama  541:    @code{sm1.gb_d} returns the results by a list of distributed polynomials.
1.1       takayama  542:     Monomials in each distributed polynomial are ordered in the given order.
                    543:     The return value consists of
                    544:     [variable names, order matrix, grobner basis in districuted polynomials,
                    545:      initial monomials or initial polynomials].
                    546: @item
                    547:    When a non-term order is given, the Grobner basis is computed in
                    548:    the homogenized Weyl algebra  (See Section 1.2 of the book of SST).
                    549:    The homogenization variable h is automatically added.
1.2       takayama  550: @item
1.8       takayama  551:    When the optional variable @var{q} is set, @code{sm1.gb} returns,
1.2       takayama  552:    as the third return value, a list of
                    553:    the Grobner basis and the initial ideal
                    554:    with sums of monomials sorted by the given order.
                    555:    Each polynomial is expressed as a string temporally for now.
1.3       takayama  556:    When the optional variable @var{r} is set to one,
                    557:    the polynomials are dehomogenized (,i.e., h is set to 1).
1.1       takayama  558: @end itemize
                    559: */
                    560: /*&jp-texi
1.8       takayama  561: @c sort-sm1.gb
                    562: @node sm1.gb,,, SM1 Functions
                    563: @node sm1.gb_d,,, SM1 Functions
                    564: @subsection @code{sm1.gb}
                    565: @findex sm1.gb
                    566: @findex sm1.gb_d
1.1       takayama  567: @table @t
1.8       takayama  568: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
1.1       takayama  569: ::  @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
1.8       takayama  570: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1       takayama  571: ::  @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
                    572: @end table
                    573:
                    574: @table @var
                    575: @item return
                    576: $B%j%9%H(B
1.3       takayama  577: @item p, q, r
1.1       takayama  578: $B?t(B
                    579: @item f, v, w
                    580: $B%j%9%H(B
                    581: @end table
                    582:
                    583: @itemize @bullet
                    584: @item
                    585:    @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
                    586: @item
                    587:    Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
                    588:    $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
                    589:    $B%V%l%V%J4pDl$r7W;;$9$k(B.
                    590: @item
1.8       takayama  591:    @code{sm1.gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B
1.1       takayama  592:   ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
                    593:   $B$N%j%9%H$G$"$k(B.
                    594: @item
1.8       takayama  595:    @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
1.1       takayama  596:     $BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.
                    597:    $BLa$jCM$O(B
                    598:     [$BJQ?tL>$N%j%9%H(B, $B=g=x$r$-$a$k9TNs(B, $B%0%l%V%J4pDl(B, $B%$%K%7%c%k%b%N%_%"%k$^$?$O%$%K%7%!%kB?9`<0(B]
                    599:    $B$G$"$k(B.
                    600: @item
                    601:    Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
                    602: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1.2       takayama  603: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
                    604:     3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
                    605:     $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
                    606:     $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
1.3       takayama  607:     $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
                    608:     $BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B).
1.1       takayama  609: @end itemize
                    610: */
                    611: /*&C-texi
                    612: @example
1.8       takayama  613: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1.1       takayama  614: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
                    615: @end example
                    616: */
                    617: /*&eg-texi
                    618: In the example above,
                    619: @tex the set $\{ x \partial_x + y \partial_y -1,
                    620:                  y^2 \partial_y^2+2\}$
                    621: is the Gr\"obner basis of the input with respect to the
                    622: graded reverse lexicographic order such that
                    623: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
                    624: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
                    625: (the initial monominals) of the Gr\"obner basis.
                    626: @end tex
                    627: */
                    628: /*&jp-texi
                    629: $B>e$NNc$K$*$$$F(B,
                    630: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
                    631:                  y^2 \partial_y^2+2\}$
                    632: $B$O(B
                    633: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
                    634: $B$G$"$k$h$&$J(B
                    635: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
                    636: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
                    637: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
                    638: @end tex
                    639: */
                    640: /*&C-texi
                    641: @example
1.8       takayama  642: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
1.1       takayama  643: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
                    644: @end example
                    645: */
                    646: /*&eg-texi
                    647: In the example above, two monomials
                    648: @tex
                    649: $m = x^a y^b \partial_x^c \partial_y^d$ and
                    650: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
                    651: are firstly compared by the weight vector
                    652: {\tt (dx,dy,x,y) = (50,2,1,0)}
                    653: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
                    654: and when the comparison is tie, then these are
                    655: compared by the reverse lexicographic order
                    656: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
                    657: @end tex
                    658: */
                    659: /*&jp-texi
                    660: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
                    661: @tex
                    662: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
                    663: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
                    664: $B$O:G=i$K(B weight vector
                    665: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
                    666: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
                    667:  $m'$ $B$h$jBg$-$$(B )
                    668: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
                    669: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
                    670: $B$5$l$k(B).
                    671: @end tex
1.2       takayama  672: */
                    673: /*&C-texi
                    674: @example
1.8       takayama  675: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
1.2       takayama  676:       map(print,F[2][0])$
                    677:       map(print,F[2][1])$
                    678: @end example
1.1       takayama  679: */
                    680: /*&C-texi
                    681: @example
                    682: [595]
1.8       takayama  683:    sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
1.1       takayama  684:              [x,y],[[dx,1,x,-1],[dy,1]]]);
                    685:
                    686: [[x*dx^2+(y*dy-h^2)*dx-h^3,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx-h^3*dy],
                    687:  [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
                    688:
                    689: [596]
1.8       takayama  690:    sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
1.1       takayama  691:              "x,y",[[dx,1,x,-1],[dy,1]]]);
                    692: [[[e0,x,y,H,E,dx,dy,h],
                    693:  [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
                    694:   [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
                    695:   [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
                    696:   [0,0,0,0,0,0,0,1]]],
                    697: [[(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>+(-1)*
                    698: <<0,0,0,0,0,0,0,3>>,(1)*<<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0
                    699: ,0,0,0,1,2>>+(-1)*<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>+(-1)*<<0,0,0,0,0,0
                    700: ,1,3>>],
                    701:  [(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>,(1)*<
                    702: <0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0,0,0,0,1,2>>+(-1)*<<0,0,0
                    703: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
                    704: @end example
                    705: */
                    706:
                    707: /*&eg-texi
                    708: @table @t
                    709: @item Reference
1.8       takayama  710:     @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1       takayama  711: @end table
                    712: */
                    713: /*&jp-texi
                    714: @table @t
                    715: @item $B;2>H(B
1.8       takayama  716:     @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1       takayama  717: @end table
                    718: */
                    719:
                    720:
                    721:
                    722: /*&eg-texi
1.8       takayama  723: @c sort-sm1.deRham
                    724: @node sm1.deRham,,, SM1 Functions
                    725: @subsection @code{sm1.deRham}
                    726: @findex sm1.deRham
1.1       takayama  727: @table @t
1.8       takayama  728: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
1.1       takayama  729: ::  ask the server to evaluate the dimensions of the de Rham cohomology  groups
                    730: of C^n - (the zero set of @var{f}=0).
                    731: @end table
                    732:
                    733: @table @var
                    734: @item return
                    735: List
                    736: @item p
                    737: Number
                    738: @item f
                    739: String or polynomial
                    740: @item v
                    741: List
                    742: @end table
                    743:
                    744: @itemize @bullet
                    745: @item  It returns the dimensions of the de Rham cohomology groups
                    746:     of X = C^n \ V(@var{f}).
                    747:    In other words,  it returns
                    748:       [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
                    749: @item   @var{v} is a list of variables. n = @code{length(@var{v})}.
                    750: @item
1.8       takayama  751:    @code{sm1.deRham} requires huge computer resources.
                    752:     For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
1.1       takayama  753:     is already very hard.
                    754: @item
                    755:  To efficiently analyze the roots of b-function, @code{ox_asir} should be used
                    756:   from @code{ox_sm1_forAsir}.
                    757:     It is recommended to load the communication module for @code{ox_asir}
                    758:     by the command @*
                    759:    @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
                    760:  This command is automatically executed when @code{ox_sm1_forAsir} is started.
1.8       takayama  761: @item If you make an interruption to the function @code{sm1.deRham}
                    762: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
1.1       takayama  763: mode. So, it is strongly recommended to execute the command
1.8       takayama  764: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
1.1       takayama  765: @end itemize
                    766: */
                    767: /*&jp-texi
1.8       takayama  768: @c sort-sm1.deRham
                    769: @node sm1.deRham,,, SM1 Functions
                    770: @subsection @code{sm1.deRham}
                    771: @findex sm1.deRham
1.1       takayama  772: @table @t
1.8       takayama  773: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
1.1       takayama  774: ::  $B6u4V(B C^n - (the zero set of @var{f}=0) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$7$F$/$l$k$h$&$K%5!<%P$KMj$`(B.
                    775: @end table
                    776:
                    777: @table @var
                    778: @item return
                    779: $B%j%9%H(B
                    780: @item p
                    781: $B?t(B
                    782: @item f
                    783: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
                    784: @item v
                    785: $B%j%9%H(B
                    786: @end table
                    787:
                    788: @itemize @bullet
                    789: @item $B$3$NH!?t$O6u4V(B X = C^n \ V(@var{f}) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$9$k(B.
                    790:    $B$9$J$o$A(B,
                    791:    [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
                    792:    $B$rLa$9(B.
                    793: @item   @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
                    794: @item
1.8       takayama  795:    @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
                    796:     $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
1.1       takayama  797:    $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
                    798: @item
                    799:   b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
                    800:   $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B.  $B%3%^%s%I(B @*
                    801:    @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
                    802:    $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
                    803:    $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
                    804: @item
1.8       takayama  805:   @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
1.1       takayama  806:   $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
1.8       takayama  807:   $B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
1.1       takayama  808:   $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
                    809: @end itemize
                    810: */
                    811: /*&C-texi
                    812: @example
1.8       takayama  813: [332] sm1.deRham([x^3-y^2,[x,y]]);
1.1       takayama  814: [1,1,0]
1.8       takayama  815: [333] sm1.deRham([x*(x-1),[x]]);
1.1       takayama  816: [1,2]
                    817: @end example
                    818: */
                    819: /*&eg-texi
                    820: @table @t
                    821: @item Reference
1.8       takayama  822:     @code{sm1.start}, @code{deRham} (sm1 command)
1.5       takayama  823: @item Algorithm:
1.1       takayama  824:     Oaku, Takayama, An algorithm for de Rham cohomology groups of the
                    825:     complement of an affine variety via D-module computation,
                    826:     Journal of pure and applied algebra 139 (1999), 201--233.
                    827: @end table
                    828: */
                    829: /*&jp-texi
                    830: @table @t
                    831: @item $B;2>H(B
1.8       takayama  832:     @code{sm1.start}, @code{deRham} (sm1 command)
1.5       takayama  833: @item Algorithm:
1.1       takayama  834:     Oaku, Takayama, An algorithm for de Rham cohomology groups of the
                    835:     complement of an affine variety via D-module computation,
                    836:     Journal of pure and applied algebra 139 (1999), 201--233.
                    837: @end table
                    838: */
                    839:
                    840:
                    841:
                    842:
                    843: /*&eg-texi
1.8       takayama  844: @c sort-sm1.hilbert
                    845: @node sm1.hilbert,,, SM1 Functions
                    846: @subsection @code{sm1.hilbert}
                    847: @findex sm1.hilbert
1.1       takayama  848: @findex hilbert_polynomial
                    849: @table @t
1.8       takayama  850: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
1.1       takayama  851: ::  ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
                    852: @item hilbert_polynomial(@var{f},@var{v})
                    853: ::  ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
                    854: @end table
                    855:
                    856: @table @var
                    857: @item return
                    858: Polynomial
                    859: @item p
                    860: Number
                    861: @item f, v
                    862: List
                    863: @end table
                    864:
                    865: @itemize @bullet
                    866: @item  It returns the Hilbert polynomial h(k) of the set of polynomials
                    867:     @var{f}
                    868:     with respect to the set of variables @var{v}.
                    869: @item
                    870:     h(k) = dim_Q F_k/I \cap F_k  where F_k the set of polynomials of which
                    871:     degree is less than or equal to k and I is the ideal generated by the
                    872:     set of polynomials @var{f}.
                    873: @item
1.8       takayama  874:    Note for sm1.hilbert:
1.1       takayama  875:    For an efficient computation, it is preferable that
                    876:    the set of polynomials @var{f} is a set of monomials.
                    877:    In fact, this function firstly compute a Grobner basis of @var{f}, and then
                    878:    compute the Hilbert polynomial of the initial monomials of the basis.
                    879:    If the input @var{f} is already a Grobner
                    880:    basis, a Grobner basis is recomputed in this function,
                    881:    which is a waste of time and Grobner basis computation in the ring of
                    882:    polynomials in @code{sm1} is  slower than in @code{asir}.
                    883: @end itemize
                    884: */
                    885: /*&jp-texi
1.8       takayama  886: @c sort-sm1.hilbert
                    887: @node sm1.hilbert,,, SM1 Functions
                    888: @subsection @code{sm1.hilbert}
                    889: @findex sm1.hilbert
1.1       takayama  890: @findex hilbert_polynomial
                    891: @table @t
1.8       takayama  892: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
1.1       takayama  893: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
                    894: @item hilbert_polynomial(@var{f},@var{v})
                    895: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
                    896: @end table
                    897:
                    898: @table @var
                    899: @item return
                    900: $BB?9`<0(B
                    901: @item p
                    902: $B?t(B
                    903: @item f, v
                    904: $B%j%9%H(B
                    905: @end table
                    906:
                    907: @itemize @bullet
                    908: @item  $BB?9`<0$N=89g(B @var{f} $B$NJQ?t(B @var{v} $B$K$+$s$9$k%R%k%Y%k%HB?9`<0(B h(k)
                    909:    $B$r7W;;$9$k(B.
                    910: @item
                    911:     h(k) = dim_Q F_k/I \cap F_k  $B$3$3$G(B F_k $B$O<!?t$,(B k $B0J2<$G$"$k$h$&$J(B
                    912:     $BB?9`<0$N=89g$G$"$k(B. I $B$OB?9`<0$N=89g(B @var{f} $B$G@8@.$5$l$k%$%G%"%k$G$"$k(B.
                    913: @item
1.8       takayama  914:    sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
1.1       takayama  915:    $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
                    916:    $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
                    917:    monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
                    918:   $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
                    919:    $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
                    920:   $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
                    921: @end itemize
                    922: */
                    923:
                    924: /*&C-texi
                    925: @example
                    926:
                    927: [346] load("katsura")$
                    928: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
                    929: 32
                    930:
                    931: @end example
                    932:
                    933: @example
                    934: [279] load("katsura")$
                    935: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
                    936: [281] dp_ord();
                    937: 0
                    938: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
                    939: [(1)*<<1,0,0,0,0,0>>,(1)*<<0,0,0,2,0,0>>,(1)*<<0,0,1,1,0,0>>,(1)*<<0,0,2,0,0,0>>,
                    940:  (1)*<<0,1,1,0,0,0>>,(1)*<<0,2,0,0,0,0>>,(1)*<<0,0,0,1,1,1>>,(1)*<<0,0,0,1,2,0>>,
                    941:  (1)*<<0,0,1,0,2,0>>,(1)*<<0,1,0,0,2,0>>,(1)*<<0,1,0,1,1,0>>,(1)*<<0,0,0,0,2,2>>,
                    942:   (1)*<<0,0,1,0,1,2>>,(1)*<<0,1,0,0,1,2>>,(1)*<<0,1,0,1,0,2>>,(1)*<<0,0,0,0,3,1>>,
                    943:   (1)*<<0,0,0,0,4,0>>,(1)*<<0,0,0,0,1,4>>,(1)*<<0,0,0,1,0,4>>,(1)*<<0,0,1,0,0,4>>,
                    944:  (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
                    945: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
                    946: [u0,u3^2,u3*u2,u2^2,u2*u1,u1^2,u5*u4*u3,u4^2*u3,u4^2*u2,u4^2*u1,u4*u3*u1,
                    947:  u5^2*u4^2,u5^2*u4*u2,u5^2*u4*u1,u5^2*u3*u1,u5*u4^3,u4^4,u5^4*u4,u5^4*u3,
                    948:  u5^4*u2,u5^4*u1,u5^6]
1.8       takayama  949: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
1.1       takayama  950: 32
                    951: @end example
                    952: */
                    953:
                    954: /*&eg-texi
                    955: @table @t
                    956: @item Reference
1.8       takayama  957:     @code{sm1.start}, @code{sm1.gb}, @code{longname}
1.1       takayama  958: @end table
                    959: */
                    960: /*&jp-texi
                    961: @table @t
                    962: @item $B;2>H(B
1.8       takayama  963:     @code{sm1.start}, @code{sm1.gb}, @code{longname}
1.1       takayama  964: @end table
                    965: */
                    966:
                    967:
                    968: /*&eg-texi
1.8       takayama  969: @c sort-sm1.genericAnn
                    970: @node sm1.genericAnn,,, SM1 Functions
                    971: @subsection @code{sm1.genericAnn}
                    972: @findex sm1.genericAnn
1.1       takayama  973: @table @t
1.8       takayama  974: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1.1       takayama  975: ::  It computes  the annihilating ideal for @var{f}^s.
                    976:     @var{v} is the list of variables.  Here, s is @var{v}[0] and
                    977:     @var{f} is a polynomial in the variables @code{rest}(@var{v}).
                    978: @end table
                    979:
                    980: @table @var
                    981: @item return
                    982: List
                    983: @item p
                    984: Number
                    985: @item f
                    986: Polynomial
                    987: @item v
                    988: List
                    989: @end table
                    990:
                    991: @itemize @bullet
                    992: @item  This function computes  the annihilating ideal for @var{f}^s.
                    993:     @var{v} is the list of variables.  Here, s is @var{v}[0] and
                    994:     @var{f} is a polynomial in the variables @code{rest}(@var{v}).
                    995: @end itemize
                    996: */
                    997: /*&jp-texi
1.8       takayama  998: @c sort-sm1.genericAnn
                    999: @node sm1.genericAnn,,, SM1 Functions
                   1000: @subsection @code{sm1.genericAnn}
                   1001: @findex sm1.genericAnn
1.1       takayama 1002: @table @t
1.8       takayama 1003: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1.1       takayama 1004: ::  @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
                   1005:     @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.  $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,
                   1006:     @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
                   1007: @end table
                   1008:
                   1009: @table @var
                   1010: @item return
                   1011: $B%j%9%H(B
                   1012: @item p
                   1013: $B?t(B
                   1014: @item f
                   1015: $BB?9`<0(B
                   1016: @item v
                   1017: $B%j%9%H(B
                   1018: @end table
                   1019:
                   1020: @itemize @bullet
                   1021: @item $B$3$NH!?t$O(B,
                   1022:   @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
                   1023:     @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.  $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,
                   1024:     @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
                   1025: @end itemize
                   1026: */
                   1027: /*&C-texi
                   1028: @example
1.8       takayama 1029: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1.1       takayama 1030: [-x*dx-y*dy-z*dz+3*s,z^2*dy-y^2*dz,z^2*dx-x^2*dz,y^2*dx-x^2*dy]
                   1031: @end example
                   1032: */
                   1033: /*&eg-texi
                   1034: @table @t
                   1035: @item Reference
1.8       takayama 1036:     @code{sm1.start}
1.1       takayama 1037: @end table
                   1038: */
                   1039: /*&jp-texi
                   1040: @table @t
                   1041: @item $B;2>H(B
1.8       takayama 1042:     @code{sm1.start}
1.1       takayama 1043: @end table
                   1044: */
                   1045:
                   1046:
                   1047:
                   1048: /*&eg-texi
1.8       takayama 1049: @c sort-sm1.wTensor0
                   1050: @node sm1.wTensor0,,, SM1 Functions
                   1051: @subsection @code{sm1.wTensor0}
                   1052: @findex sm1.wTensor0
1.1       takayama 1053: @table @t
1.8       takayama 1054: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1       takayama 1055: ::   It computes the D-module theoretic 0-th tensor product
                   1056:     of @var{f} and @var{g}.
                   1057: @end table
                   1058:
                   1059: @table @var
                   1060: @item return
                   1061: List
                   1062: @item p
                   1063: Number
                   1064: @item f, g, v, w
                   1065: List
                   1066: @end table
                   1067:
                   1068: @itemize @bullet
                   1069: @item
                   1070:    It returns the D-module theoretic 0-th tensor product
                   1071:    of @var{f} and @var{g}.
                   1072: @item
                   1073:   @var{v} is a list of variables.
                   1074:   @var{w} is a list of weights.  The integer @var{w}[i] is
                   1075:   the weight of the variable @var{v}[i].
                   1076: @item
1.8       takayama 1077:    @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1.1       takayama 1078:    which requires a generic weight
                   1079:     vector @var{w} to compute the restriction.
                   1080:     If @var{w} is not generic, the computation fails.
                   1081: @item Let F and G be solutions of @var{f} and @var{g} respectively.
                   1082: Intuitively speaking, the 0-th tensor product is a system of
                   1083: differential equations which annihilates the function FG.
                   1084: @item The answer is a submodule of a free module D^r in general even if
                   1085: the inputs @var{f} and @var{g} are left ideals of D.
                   1086: @end itemize
                   1087: */
                   1088:
                   1089: /*&jp-texi
1.8       takayama 1090: @c sort-sm1.wTensor0
                   1091: @node sm1.wTensor0,,, SM1 Functions
                   1092: @subsection @code{sm1.wTensor0}
                   1093: @findex sm1.wTensor0
1.1       takayama 1094: @table @t
1.8       takayama 1095: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1       takayama 1096: ::   @var{f} $B$H(B @var{g} $B$N(B D-module $B$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r(B
                   1097: $B7W;;$9$k(B.
                   1098: @end table
                   1099:
                   1100: @table @var
                   1101: @item return
                   1102: $B%j%9%H(B
                   1103: @item p
                   1104: $B?t(B
                   1105: @item f, g, v, w
                   1106: $B%j%9%H(B
                   1107: @end table
                   1108:
                   1109: @itemize @bullet
                   1110: @item
                   1111:    @var{f} $B$H(B @var{g} $B$N(B
                   1112:    D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
                   1113: @item
                   1114:   @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
                   1115:   @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
                   1116:   $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
                   1117: @item
1.8       takayama 1118:    @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1.1       takayama 1119:    $B$r$h$s$G$$$k(B.
                   1120:   @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
                   1121:   $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
                   1122:   Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
                   1123: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B  @var{g} $B$=$l$>$l$N2r$H$9$k(B.
                   1124: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
                   1125: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
                   1126: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
                   1127: @end itemize
                   1128: */
                   1129: /*&C-texi
                   1130: @example
1.8       takayama 1131: [258]  sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1.1       takayama 1132: [[-y*x*dx-y*x*dy+4*x+y],[5*x*dx^2+5*x*dx+2*y*dy^2+(-2*y-6)*dy+3],
                   1133:  [-25*x*dx+(-5*y*x-2*y^2)*dy^2+((5*y+15)*x+2*y^2+16*y)*dy-20*x-8*y-15],
                   1134:  [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
                   1135: @end example
                   1136: */
                   1137:
                   1138:
                   1139:
                   1140: /*&eg-texi
1.8       takayama 1141: @c sort-sm1.reduction
                   1142: @node sm1.reduction,,, SM1 Functions
                   1143: @subsection @code{sm1.reduction}
                   1144: @findex sm1.reduction
1.1       takayama 1145: @table @t
1.8       takayama 1146: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1       takayama 1147: ::
                   1148: @end table
                   1149:
                   1150: @table @var
                   1151: @item return
                   1152: List
                   1153: @item f
                   1154: Polynomial
                   1155: @item g, v, w
                   1156: List
                   1157: @item p
                   1158: Number  (the process number of ox_sm1)
                   1159: @end table
                   1160:
                   1161: @itemize @bullet
                   1162: @item  It reduces @var{f} by the set of polynomial @var{g}
                   1163: in the homogenized Weyl algebra; it applies the
                   1164: division algorithm to @var{f}. The set of variables is @var{v} and
                   1165: @var{w} is weight vectors to determine the order, which can be ommited.
1.8       takayama 1166: @code{sm1.reduction_noH} is for the Weyl algebra.
1.1       takayama 1167: @item The return value is of the form
                   1168: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1.7       takayama 1169: c0 f + c1 g1 + ... + cm gm = r.
1.1       takayama 1170: r/c0 is the normal form.
                   1171: @item The function reduction reduces reducible terms that appear
                   1172: in lower order terms.
                   1173: @item  The functions
1.8       takayama 1174: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1.1       takayama 1175: are for distributed polynomials.
                   1176: @end itemize
                   1177: */
                   1178: /*&jp-texi
1.8       takayama 1179: @node sm1.reduction,,, SM1 Functions
                   1180: @subsection @code{sm1.reduction}
                   1181: @findex sm1.reduction
1.1       takayama 1182: @table @t
1.8       takayama 1183: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1       takayama 1184: ::
                   1185: @end table
                   1186:
                   1187: @table @var
                   1188: @item return
                   1189: $B%j%9%H(B
                   1190: @item f
                   1191: $BB?9`<0(B
                   1192: @item g, v, w
                   1193: $B%j%9%H(B
                   1194: @item p
                   1195: $B?t(B  (ox_sm1 $B$N%W%m%;%9HV9f(B)
                   1196: @end table
                   1197:
                   1198: @itemize @bullet
                   1199: @item  $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
                   1200: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
                   1201: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
                   1202: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
                   1203: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
                   1204: $B>JN,$7$F$b$h$$(B.
1.8       takayama 1205: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1.1       takayama 1206: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1.7       takayama 1207: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
                   1208: c0 f + c1 g1 + ... + cm gm = r
1.1       takayama 1209: $B$,$J$j$?$D(B.
                   1210: r/c0 $B$,(B normal form $B$G$"$k(B.
                   1211: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
                   1212: @item  $BH!?t(B
1.8       takayama 1213: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1.1       takayama 1214: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
                   1215: @end itemize
                   1216: */
                   1217: /*&C-texi
                   1218: @example
1.8       takayama 1219: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1.7       takayama 1220: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1.8       takayama 1221: [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
1.7       takayama 1222: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1.1       takayama 1223: @end example
                   1224: */
                   1225: /*&eg-texi
                   1226: @table @t
                   1227: @item Reference
1.10      takayama 1228:     @code{sm1.start}, @code{d_true_nf}
1.1       takayama 1229: @end table
                   1230: */
                   1231: /*&jp-texi
                   1232: @table @t
                   1233: @item $B;2>H(B
1.10      takayama 1234:     @code{sm1.start}, @code{d_true_nf}
1.1       takayama 1235: @end table
                   1236: */
                   1237:
                   1238:
                   1239: /*&eg-texi
1.8       takayama 1240: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
                   1241: @subsection @code{sm1.xml_tree_to_prefix_string}
                   1242: @findex sm1.xml_tree_to_prefix_string
1.1       takayama 1243: @table @t
1.8       takayama 1244: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1.1       takayama 1245: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
                   1246: @end table
                   1247:
                   1248: @table @var
                   1249: @item return
                   1250: String
                   1251: @item p
                   1252: Number
                   1253: @item s
                   1254: String
                   1255: @end table
                   1256:
                   1257: @itemize @bullet
                   1258: @item  It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
                   1259: @item This function should be moved to om_* in a future.
                   1260: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
                   1261: asir has not yet understood this CMO.
                   1262: @item @code{java} execution environment is required.
                   1263: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
                   1264: command search path.)
                   1265: @end itemize
                   1266: */
                   1267: /*&jp-texi
1.8       takayama 1268: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
                   1269: @subsection @code{sm1.xml_tree_to_prefix_string}
                   1270: @findex sm1.xml_tree_to_prefix_string
1.1       takayama 1271: @table @t
1.8       takayama 1272: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1.1       takayama 1273: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
                   1274: @end table
                   1275:
                   1276: @table @var
                   1277: @item return
                   1278: String
                   1279: @item p
                   1280: Number
                   1281: @item s
                   1282: String
                   1283: @end table
                   1284:
                   1285: @itemize @bullet
                   1286: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
                   1287: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
                   1288: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
                   1289: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
                   1290: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
                   1291: ($B$?$H$($P(B, /usr/local/jdk1.1.8/bin $B$r%3%^%s%I%5!<%A%Q%9$KF~$l$k$J$I(B.)
                   1292: @end itemize
                   1293: */
                   1294: /*&C-texi
                   1295: @example
                   1296: [263] load("om");
                   1297: 1
                   1298: [270] F=om_xml(x^4-1);
                   1299: control: wait OX
                   1300: Trying to connect to the server... Done.
                   1301: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
                   1302: <OMS name="times" cd="basic"/><OMA>
                   1303: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
                   1304: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
                   1305: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
                   1306: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1.8       takayama 1307: [271] sm1.xml_tree_to_prefix_string(F);
1.1       takayama 1308: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
                   1309: @end example
                   1310: */
                   1311: /*&eg-texi
                   1312: @table @t
                   1313: @item Reference
                   1314:     @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
                   1315: @end table
                   1316: */
                   1317: /*&jp-texi
                   1318: @table @t
                   1319: @item $B;2>H(B
                   1320:     @code{om_*}, @code{OpenXM/src/OpenMath},  @code{eval_str}
                   1321: @end table
                   1322: */
                   1323:
                   1324:
                   1325:
                   1326:
                   1327: /*&eg-texi
1.8       takayama 1328: @c sort-sm1.syz
                   1329: @node sm1.syz,,, SM1 Functions
                   1330: @node sm1.syz_d,,, SM1 Functions
                   1331: @subsection @code{sm1.syz}
                   1332: @findex sm1.syz
                   1333: @findex sm1.syz_d
1.1       takayama 1334: @table @t
1.8       takayama 1335: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1       takayama 1336: ::  computes the syzygy of @var{f} in the ring of differential
                   1337: operators with the variable @var{v}.
                   1338: @end table
                   1339:
                   1340: @table @var
                   1341: @item return
                   1342: List
                   1343: @item p
                   1344: Number
                   1345: @item f, v, w
                   1346: List
                   1347: @end table
                   1348:
                   1349: @itemize @bullet
                   1350: @item
                   1351: The return values is of the form
                   1352: [@var{s},[@var{g}, @var{m}, @var{t}]].
                   1353: Here @var{s} is the syzygy of @var{f} in the ring of differential
                   1354: operators with the variable @var{v}.
                   1355: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
                   1356: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
                   1357: basis @var {g}.
                   1358: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
                   1359: In summary, @var{g} = @var{m} @var{f} and
                   1360: @var{s} @var{f} = 0 hold as matrices.
                   1361: @item
                   1362:    The weight vectors are given by @var{w}, which can be omitted.
                   1363:     If @var{w} is not given,
                   1364:     the graded reverse lexicographic order will be used to compute Grobner basis.
                   1365: @item
                   1366:    When a non-term order is given, the Grobner basis is computed in
                   1367:    the homogenized Weyl algebra  (See Section 1.2 of the book of SST).
                   1368:    The homogenization variable h is automatically added.
                   1369: @end itemize
                   1370: */
                   1371: /*&jp-texi
1.8       takayama 1372: @c sort-sm1.syz
                   1373: @node sm1.syz,,, SM1 Functions
                   1374: @node sm1.syz_d,,, SM1 Functions
                   1375: @subsection @code{sm1.syz}
                   1376: @findex sm1.syz
                   1377: @findex sm1.syz_d
1.1       takayama 1378: @table @t
1.8       takayama 1379: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1       takayama 1380: ::  @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
                   1381: @end table
                   1382:
                   1383: @table @var
                   1384: @item return
                   1385: $B%j%9%H(B
                   1386: @item p
                   1387: $B?t(B
                   1388: @item f, v, w
                   1389: $B%j%9%H(B
                   1390: @end table
                   1391:
                   1392: @itemize @bullet
                   1393: @item
                   1394: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
                   1395: [@var{s},[@var{g}, @var{m}, @var{t}]].
                   1396: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
                   1397: syzygy $B$G$"$k(B.
                   1398: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
                   1399: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
                   1400: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
                   1401: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
                   1402: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
                   1403: @var{g} = @var{m} @var{f} ,
                   1404: @var{s} @var{f} = 0.
                   1405: @item
                   1406:    Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
                   1407:    $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
                   1408:    $B%V%l%V%J4pDl$r7W;;$9$k(B.
                   1409: @item
                   1410:    Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
                   1411: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
                   1412: @end itemize
                   1413: */
                   1414: /*&C-texi
                   1415: @example
1.8       takayama 1416: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1.1       takayama 1417: [[[y*x*dy*dx-2,-x*dx-y*dy+1]],    generators of the syzygy
                   1418:  [[[x*dx+y*dy-1],[y^2*dy^2+2]],   grobner basis
                   1419:   [[1,0],[y*dy,-1]],              transformation matrix
                   1420:  [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
                   1421: @end example
                   1422: */
                   1423: /*&C-texi
                   1424: @example
1.8       takayama 1425: [294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);
1.1       takayama 1426: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
                   1427:  [[[x^2*dx^2+h^2*x*dx+y^2*dy^2+h^2*y*dy-4*h^4],[y*x*dy*dx-h^4], GB
                   1428:   [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
                   1429:  [[1,0],[0,1],[y*dy,-x*dx]],     transformation matrix
                   1430:  [[y*x*dy*dx-h^4,-x^2*dx^2-h^2*x*dx-y^2*dy^2-h^2*y*dy+4*h^4]]]]
                   1431: @end example
                   1432: */
                   1433:
                   1434:
                   1435:
                   1436: /*&eg-texi
1.8       takayama 1437: @node sm1.mul,,, SM1 Functions
                   1438: @subsection @code{sm1.mul}
                   1439: @findex sm1.mul
1.1       takayama 1440: @table @t
1.8       takayama 1441: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1.1       takayama 1442: ::  ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
                   1443: @end table
                   1444:
                   1445: @table @var
                   1446: @item return
                   1447: Polynomial or List
                   1448: @item p
                   1449: Number
                   1450: @item f, g
                   1451: Polynomial or List
                   1452: @item v
                   1453: List
                   1454: @end table
                   1455:
                   1456: @itemize @bullet
                   1457: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1.8       takayama 1458: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1.1       takayama 1459: @end itemize
                   1460: */
                   1461:
                   1462: /*&jp-texi
1.8       takayama 1463: @node sm1.mul,,, SM1 Functions
                   1464: @subsection @code{sm1.mul}
                   1465: @findex sm1.mul
1.1       takayama 1466: @table @t
1.8       takayama 1467: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1.1       takayama 1468: ::  sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
                   1469: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
                   1470: @end table
                   1471:
                   1472: @table @var
                   1473: @item return
                   1474: $BB?9`<0$^$?$O%j%9%H(B
                   1475: @item p
                   1476: $B?t(B
                   1477: @item f, g
                   1478: $BB?9`<0$^$?$O%j%9%H(B
                   1479: @item v
                   1480: $B%j%9%H(B
                   1481: @end table
                   1482:
                   1483: @itemize @bullet
                   1484: @item   sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
                   1485: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1.8       takayama 1486: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1.1       takayama 1487: @end itemize
                   1488: */
                   1489:
                   1490: /*&C-texi
                   1491:
                   1492: @example
1.8       takayama 1493: [277] sm1.mul(dx,x,[x]);
1.1       takayama 1494: x*dx+1
1.8       takayama 1495: [278] sm1.mul([x,y],[1,2],[x,y]);
1.1       takayama 1496: x+2*y
1.8       takayama 1497: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1.1       takayama 1498: [[x+2,y+4],[3*x+4,3*y+8]]
                   1499: @end example
                   1500:
                   1501: */
                   1502:
                   1503:
                   1504:
                   1505:
                   1506: /*&eg-texi
1.8       takayama 1507: @node sm1.distraction,,, SM1 Functions
                   1508: @subsection @code{sm1.distraction}
                   1509: @findex sm1.distraction
1.1       takayama 1510: @table @t
1.8       takayama 1511: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1.1       takayama 1512: ::  ask the @code{sm1} server to compute the distraction of @var{f}.
                   1513: @end table
                   1514:
                   1515: @table @var
                   1516: @item return
                   1517: List
                   1518: @item p
                   1519: Number
                   1520: @item f
                   1521: Polynomial
                   1522: @item v,x,d,s
                   1523: List
                   1524: @end table
                   1525:
                   1526: @itemize @bullet
                   1527: @item  It asks the @code{sm1} server of the descriptor number @var{p}
                   1528: to compute the distraction of  @var{f} in the ring of differential
                   1529: operators with variables @var{v}.
                   1530: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
                   1531: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
                   1532: @item Distraction is roughly speaking to replace x*dx by a single variable x.
                   1533: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
                   1534: @end itemize
                   1535: */
                   1536:
                   1537: /*&jp-texi
1.8       takayama 1538: @node sm1.distraction,,, SM1 Functions
1.1       takayama 1539:
1.8       takayama 1540: @subsection @code{sm1.distraction}
                   1541: @findex sm1.distraction
1.1       takayama 1542: @table @t
1.8       takayama 1543: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1.1       takayama 1544: ::  @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
                   1545: @end table
                   1546:
                   1547: @table @var
                   1548: @item return
                   1549: $B%j%9%H(B
                   1550: @item p
                   1551: $B?t(B
                   1552: @item f
                   1553: $BB?9`<0(B
                   1554: @item v,x,d,s
                   1555: $B%j%9%H(B
                   1556: @end table
                   1557:
                   1558: @itemize @bullet
                   1559: @item  $B<1JL;R(B @var{p}  $B$N(B @code{sm1}  $B%5!<%P$K(B,
                   1560: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
                   1561: @item @var{x} , @var{d} $B$O(B, $B$=$l$>$l(B, distract $B$9$Y$-(B x $BJQ?t(B, d $BJQ?t$N(B
                   1562: $B%j%9%H(B.  Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
                   1563: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
                   1564: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
                   1565: @end itemize
                   1566: */
                   1567:
                   1568: /*&C-texi
                   1569:
                   1570: @example
1.8       takayama 1571: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1.1       takayama 1572: x
1.8       takayama 1573: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1.1       takayama 1574: x^2-x
1.8       takayama 1575: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1.1       takayama 1576: x^2+3*x+2
                   1577: [283] fctr(@@);
                   1578: [[1,1],[x+1,1],[x+2,1]]
1.8       takayama 1579: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1.1       takayama 1580: (x^2-x)*dy+x*y
                   1581: @end example
                   1582: */
                   1583:
                   1584: /*&eg-texi
                   1585: @table @t
                   1586: @item Reference
                   1587:     @code{distraction2(sm1)},
                   1588: @end table
                   1589: */
                   1590:
                   1591: /*&jp-texi
                   1592: @table @t
                   1593: @item $B;2>H(B
                   1594:     @code{distraction2(sm1)},
                   1595: @end table
                   1596: */
                   1597:
                   1598:
                   1599:
                   1600: /*&eg-texi
1.8       takayama 1601: @node sm1.gkz,,, SM1 Functions
                   1602: @subsection @code{sm1.gkz}
                   1603: @findex sm1.gkz
1.1       takayama 1604: @table @t
1.8       takayama 1605: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1.1       takayama 1606: ::  Returns the GKZ system (A-hypergeometric system) associated to the matrix
                   1607: @var{A} with the parameter vector @var{B}.
                   1608: @end table
                   1609:
                   1610: @table @var
                   1611: @item return
                   1612: List
                   1613: @item p
                   1614: Number
                   1615: @item A, B
                   1616: List
                   1617: @end table
                   1618:
                   1619: @itemize @bullet
                   1620: @item Returns the GKZ hypergeometric system
                   1621: (A-hypergeometric system) associated to the matrix
                   1622: @end itemize
                   1623: */
                   1624:
                   1625: /*&jp-texi
1.8       takayama 1626: @node sm1.gkz,,, SM1 Functions
                   1627: @subsection @code{sm1.gkz}
                   1628: @findex sm1.gkz
1.1       takayama 1629: @table @t
1.8       takayama 1630: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1.1       takayama 1631: ::  $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
                   1632: @end table
                   1633:
                   1634: @table @var
                   1635: @item return
                   1636: $B%j%9%H(B
                   1637: @item p
                   1638: $B?t(B
                   1639: @item A, B
                   1640: $B%j%9%H(B
                   1641: @end table
                   1642:
                   1643: @itemize @bullet
                   1644: @item  $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
                   1645: @end itemize
                   1646: */
                   1647:
                   1648: /*&C-texi
                   1649:
                   1650: @example
                   1651:
1.8       takayama 1652: [280] sm1.gkz([  [[1,1,1,1],[0,1,3,4]],  [0,2] ]);
1.1       takayama 1653: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
                   1654:  -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
                   1655:  [x1,x2,x3,x4]]
                   1656:
                   1657: @end example
                   1658:
                   1659: */
                   1660:
                   1661:
                   1662:
                   1663:
                   1664: /*&eg-texi
1.8       takayama 1665: @node sm1.appell1,,, SM1 Functions
                   1666: @subsection @code{sm1.appell1}
                   1667: @findex sm1.appell1
1.1       takayama 1668: @table @t
1.8       takayama 1669: @item sm1.appell1(@var{a}|proc=@var{p})
1.1       takayama 1670: ::  Returns the Appell hypergeometric system F_1 or F_D.
                   1671: @end table
                   1672:
                   1673: @table @var
                   1674: @item return
                   1675: List
                   1676: @item p
                   1677: Number
                   1678: @item a
                   1679: List
                   1680: @end table
                   1681:
                   1682: @itemize @bullet
                   1683: @item Returns the hypergeometric system for the Lauricella function
                   1684: F_D(a,b1,b2,...,bn,c;x1,...,xn)
                   1685: where @var{a} =(a,c,b1,...,bn).
                   1686: When n=2, the Lauricella function is called the Appell function F_1.
                   1687: The parameters a, c, b1, ..., bn may be rational numbers.
                   1688: @end itemize
                   1689: */
                   1690:
                   1691: /*&jp-texi
1.8       takayama 1692: @node sm1.appell1,,, SM1 Functions
                   1693: @subsection @code{sm1.appell1}
                   1694: @findex sm1.appell1
1.1       takayama 1695: @table @t
1.8       takayama 1696: @item sm1.appell1(@var{a}|proc=@var{p})
1.1       takayama 1697: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
                   1698: @end table
                   1699:
                   1700: @table @var
                   1701: @item return
                   1702: $B%j%9%H(B
                   1703: @item p
                   1704: $B?t(B
                   1705: @item a
                   1706: $B%j%9%H(B
                   1707: @end table
                   1708:
                   1709: @itemize @bullet
                   1710: @item Appell $B$N4X?t(B F_1 $B$*$h$S(B $B$=$N(B n $BJQ?t2=$G$"$k(B Lauricella $B$N4X?t(B
                   1711: F_D(a,b1,b2,...,bn,c;x1,...,xn)
                   1712: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
                   1713: @var{a} =(a,c,b1,...,bn).
                   1714: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
                   1715: @end itemize
                   1716: */
                   1717:
                   1718: /*&C-texi
                   1719:
                   1720: @example
                   1721:
1.8       takayama 1722: [281] sm1.appell1([1,2,3,4]);
1.1       takayama 1723: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
                   1724:   (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
                   1725:   ((-x2+x1)*dx1+3)*dx2-4*dx1],       equations
                   1726:  [x1,x2]]                            the list of variables
                   1727:
1.8       takayama 1728: [282] sm1.gb(@@);
1.1       takayama 1729: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
                   1730:   +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
                   1731:   +(-4*x2-4*x1)*dx1-4,
                   1732:   (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
                   1733:  +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
                   1734:  [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
                   1735:
1.8       takayama 1736: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1.1       takayama 1737: 1
                   1738:
                   1739: [285] Mu=2$ Beta = 1/3$
1.8       takayama 1740: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1.1       takayama 1741: 4
                   1742:
                   1743:
                   1744: @end example
                   1745:
                   1746: */
                   1747:
                   1748: /*&eg-texi
1.8       takayama 1749: @node sm1.appell4,,, SM1 Functions
                   1750: @subsection @code{sm1.appell4}
                   1751: @findex sm1.appell4
1.1       takayama 1752: @table @t
1.8       takayama 1753: @item sm1.appell4(@var{a}|proc=@var{p})
1.1       takayama 1754: ::  Returns the Appell hypergeometric system F_4 or F_C.
                   1755: @end table
                   1756:
                   1757: @table @var
                   1758: @item return
                   1759: List
                   1760: @item p
                   1761: Number
                   1762: @item a
                   1763: List
                   1764: @end table
                   1765:
                   1766: @itemize @bullet
                   1767: @item Returns the hypergeometric system for the Lauricella function
                   1768: F_4(a,b,c1,c2,...,cn;x1,...,xn)
                   1769: where @var{a} =(a,b,c1,...,cn).
                   1770: When n=2, the Lauricella function is called the Appell function F_4.
                   1771: The parameters a, b, c1, ..., cn may be rational numbers.
                   1772: @end itemize
                   1773: */
                   1774:
                   1775: /*&jp-texi
1.8       takayama 1776: @node sm1.appell4,,, SM1 Functions
                   1777: @subsection @code{sm1.appell4}
                   1778: @findex sm1.appell4
1.1       takayama 1779: @table @t
1.8       takayama 1780: @item sm1.appell4(@var{a}|proc=@var{p})
1.1       takayama 1781: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
                   1782: @end table
                   1783:
                   1784: @table @var
                   1785: @item return
                   1786: $B%j%9%H(B
                   1787: @item p
                   1788: $B?t(B
                   1789: @item a
                   1790: $B%j%9%H(B
                   1791: @end table
                   1792:
                   1793: @itemize @bullet
                   1794: @item Appell $B$N4X?t(B F_4 $B$*$h$S(B $B$=$N(B n $BJQ?t2=$G$"$k(B Lauricella $B$N4X?t(B
                   1795: F_C(a,b,c1,c2,...,cn;x1,...,xn)
                   1796: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
                   1797: @var{a} =(a,b,c1,...,cn).
                   1798: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
                   1799: @end itemize
                   1800: */
                   1801:
                   1802: /*&C-texi
                   1803:
                   1804: @example
                   1805:
1.8       takayama 1806: [281] sm1.appell4([1,2,3,4]);
1.1       takayama 1807:   [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
                   1808:   (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
                   1809:                                                               equations
                   1810:     [x1,x2]]                                      the list of variables
                   1811:
1.8       takayama 1812: [282] sm1.rank(@@);
1.1       takayama 1813: 4
                   1814:
                   1815: @end example
                   1816:
                   1817: */
                   1818:
                   1819:
                   1820:
                   1821:
                   1822: /*&eg-texi
1.8       takayama 1823: @node sm1.rank,,, SM1 Functions
                   1824: @subsection @code{sm1.rank}
                   1825: @findex sm1.rank
1.1       takayama 1826: @table @t
1.8       takayama 1827: @item sm1.rank(@var{a}|proc=@var{p})
1.1       takayama 1828: ::  Returns the holonomic rank of the system of differential equations @var{a}.
                   1829: @end table
                   1830:
                   1831: @table @var
                   1832: @item return
                   1833: Number
                   1834: @item p
                   1835: Number
                   1836: @item a
                   1837: List
                   1838: @end table
                   1839:
                   1840: @itemize @bullet
                   1841: @item It evaluates the dimension of the space of holomorphic solutions
                   1842: at a generic point of the system of differential equations @var{a}.
                   1843: The dimension is called the holonomic rank.
                   1844: @item @var{a} is a list consisting of a list of differential equations and
                   1845: a list of variables.
1.8       takayama 1846: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
                   1847: holonomic. It is generally faster than @code{sm1.rank}.
1.1       takayama 1848: @end itemize
                   1849: */
                   1850:
                   1851: /*&jp-texi
1.8       takayama 1852: @node sm1.rank,,, SM1 Functions
                   1853: @subsection @code{sm1.rank}
                   1854: @findex sm1.rank
1.1       takayama 1855: @table @t
1.8       takayama 1856: @item sm1.rank(@var{a}|proc=@var{p})
1.1       takayama 1857: ::  $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
                   1858: @end table
                   1859:
                   1860: @table @var
                   1861: @item return
                   1862: $B?t(B
                   1863: @item p
                   1864: $B?t(B
                   1865: @item a
                   1866: $B%j%9%H(B
                   1867: @end table
                   1868:
                   1869: @itemize @bullet
                   1870: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
                   1871: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
                   1872: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1.8       takayama 1873: @item  @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1.1       takayama 1874: $B$b(B holonomic rank $B$rLa$9(B.
1.8       takayama 1875: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1.1       takayama 1876: @end itemize
                   1877: */
                   1878:
                   1879: /*&C-texi
                   1880:
                   1881: @example
                   1882:
1.8       takayama 1883: [284]  sm1.gkz([  [[1,1,1,1],[0,1,3,4]],  [0,2] ]);
1.1       takayama 1884: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
                   1885:   -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
                   1886:  [x1,x2,x3,x4]]
1.8       takayama 1887: [285] sm1.rrank(@@);
1.1       takayama 1888: 4
                   1889:
1.8       takayama 1890: [286]  sm1.gkz([  [[1,1,1,1],[0,1,3,4]],  [1,2]]);
1.1       takayama 1891: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
                   1892:  -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
                   1893:  [x1,x2,x3,x4]]
1.8       takayama 1894: [287] sm1.rrank(@@);
1.1       takayama 1895: 5
                   1896:
                   1897: @end example
                   1898:
                   1899: */
                   1900:
                   1901:
                   1902: /*&eg-texi
1.8       takayama 1903: @node sm1.auto_reduce,,, SM1 Functions
                   1904: @subsection @code{sm1.auto_reduce}
                   1905: @findex sm1.auto_reduce
1.1       takayama 1906: @table @t
1.8       takayama 1907: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1.1       takayama 1908: ::  Set the flag "AutoReduce" to @var{s}.
                   1909: @end table
                   1910:
                   1911: @table @var
                   1912: @item return
                   1913: Number
                   1914: @item p
                   1915: Number
                   1916: @item s
                   1917: Number
                   1918: @end table
                   1919:
                   1920: @itemize @bullet
                   1921: @item  If @var{s} is 1, then all Grobner bases to be computed
                   1922: will be the reduced Grobner bases.
                   1923: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
                   1924: Grobner bases.  This is the default.
                   1925: @end itemize
                   1926: */
                   1927:
                   1928: /*&jp-texi
1.8       takayama 1929: @node sm1.auto_reduce,,, SM1 Functions
                   1930: @subsection @code{sm1.auto_reduce}
                   1931: @findex sm1.auto_reduce
1.1       takayama 1932: @table @t
1.8       takayama 1933: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1.1       takayama 1934: ::  $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
                   1935: @end table
                   1936:
                   1937: @table @var
                   1938: @item $BLa$jCM(B
                   1939: $B?t(B
                   1940: @item p
                   1941: $B?t(B
                   1942: @item s
                   1943: $B?t(B
                   1944: @end table
                   1945:
                   1946: @itemize @bullet
                   1947: @item  @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
                   1948: reduced $B%0%l%V%J4pDl$H$J$k(B.
                   1949: @item  @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
                   1950: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
                   1951: @end itemize
                   1952: */
                   1953:
                   1954:
                   1955:
                   1956: /*&eg-texi
1.8       takayama 1957: @node sm1.slope,,, SM1 Functions
                   1958: @subsection @code{sm1.slope}
                   1959: @findex sm1.slope
1.1       takayama 1960: @table @t
1.8       takayama 1961: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
1.1       takayama 1962: ::  Returns the slopes of differential equations @var{ii}.
                   1963: @end table
                   1964:
                   1965: @table @var
                   1966: @item return
                   1967: List
                   1968: @item p
                   1969: Number
                   1970: @item ii
                   1971: List  (equations)
                   1972: @item v
                   1973: List  (variables)
                   1974: @item f_filtration
                   1975: List  (weight vector)
                   1976: @item v_filtration
                   1977: List (weight vector)
                   1978: @end table
                   1979:
                   1980: @itemize @bullet
1.8       takayama 1981: @item @code{sm1.slope} returns the (geometric) slopes
1.1       takayama 1982: of the system of differential equations @var{ii}
                   1983: along the hyperplane specified by
                   1984: the V filtration @var{v_filtration}.
                   1985: @item @var{v} is a list of variables.
                   1986: @item The return value is a list of lists.
                   1987: The first entry of each list is the slope and the second entry
                   1988: is the weight vector for which the microcharacteristic variety is
                   1989: not bihomogeneous.
                   1990: @end itemize
1.5       takayama 1991:
                   1992: @noindent
                   1993: Algorithm:
                   1994: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
                   1995: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
                   1996: Note that the signs of the slopes are negative, but the absolute values
                   1997: of the slopes are returned.
                   1998:
1.1       takayama 1999: */
                   2000:
                   2001: /*&jp-texi
1.8       takayama 2002: @node sm1.slope,,, SM1 Functions
                   2003: @subsection @code{sm1.slope}
                   2004: @findex sm1.slope
1.1       takayama 2005: @table @t
1.8       takayama 2006: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
1.1       takayama 2007: ::  $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
                   2008: @end table
                   2009:
                   2010: @table @var
                   2011: @item return
                   2012: $B?t(B
                   2013: @item p
                   2014: $B?t(B
                   2015: @item ii
                   2016: $B%j%9%H(B  ($BJ}Dx<0(B)
                   2017: @item v
                   2018: $B%j%9%H(B ($BJQ?t(B)
                   2019: @item f_filtration
                   2020: $B%j%9%H(B  (weight vector)
                   2021: @item v_filtration
                   2022: $B%j%9%H(B (weight vector)
                   2023: @end table
                   2024:
                   2025: @itemize @bullet
1.8       takayama 2026: @item @code{sm1.slope} $B$O(B
1.1       takayama 2027: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration  @var{v_filtration}
                   2028: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
                   2029: @item @var{v} $B$OJQ?t$N%j%9%H(B.
1.5       takayama 2030: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
                   2031: $B@.J,%j%9%H$NBh(B 1 $BMWAG$,(B slope, $BBh(B 2 $BMWAG$O(B, $B$=$N(B weight vector $B$KBP1~$9$k(B
                   2032: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
                   2033: @end itemize
                   2034:
                   2035: @noindent
                   2036: Algorithm:
1.1       takayama 2037: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
                   2038: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
                   2039: $B$r$_$h(B.
                   2040: Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
                   2041: Slope $B$N@dBPCM$rLa$9(B.
                   2042: */
                   2043:
                   2044: /*&C-texi
                   2045:
                   2046: @example
                   2047:
1.8       takayama 2048: [284] A= sm1.gkz([  [[1,2,3]],  [-3] ]);
1.1       takayama 2049:
                   2050:
1.8       takayama 2051: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
1.1       takayama 2052:
1.8       takayama 2053: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
1.1       takayama 2054:      (* This is an interesting example given by Laura Matusevich,
                   2055:         June 9, 2001 *)
                   2056:
1.8       takayama 2057: [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
1.1       takayama 2058:
                   2059:
                   2060: @end example
                   2061:
                   2062: */
                   2063: /*&eg-texi
                   2064: @table @t
                   2065: @item Reference
1.10      takayama 2066:     @code{sm.gb}
1.1       takayama 2067: @end table
                   2068: */
                   2069: /*&jp-texi
                   2070: @table @t
                   2071: @item $B;2>H(B
1.10      takayama 2072:     @code{sm.gb}
1.1       takayama 2073: @end table
1.4       takayama 2074: */
                   2075:
                   2076:
                   2077: /*&eg-texi
                   2078: @include sm1-auto-en.texi
                   2079: */
                   2080:
                   2081: /*&jp-texi
                   2082: @include sm1-auto-ja.texi
1.1       takayama 2083: */
                   2084:
                   2085:
                   2086: end$
                   2087:

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