File: [local] / OpenXM / src / asir-contrib / packages / doc / sm1 / sm1.oxw (download)
Revision 1.4, Mon Jun 11 05:23:52 2012 UTC (12 years, 2 months ago) by takayama
Branch: MAIN
Changes since 1.3: +15 -4
lines
Added an explanation on slopes, constant r, and constant kappa
in the Borel and Laplace transformations respectively.
|
/*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.4 2012/06/11 05:23:52 takayama Exp $ */
/*&C
@c DO NOT EDIT THIS FILE
*/
/*&C
@node SM1 Functions,,, Top
*/
/*&ja
@chapter SM1 $BH!?t(B
$B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}
$B$H$N%$%s%?%U%'!<%94X?t$r2r@b$9$k(B.
$B$3$l$i$N4X?t$O%U%!%$%k(B @file{sm1.rr} $B$GDj5A$5$l$F$$$k(B.
@file{sm1.rr} $B$O(B @file{$(OpenXM_HOME)/lib/asir-contrib} $B$K$"$k(B.
$B%7%9%F%`(B @code{sm1} $B$OHyJ,:nMQAG4D$G7W;;$9$k$?$a$N%7%9%F%`$G$"$k(B.
$B7W;;Be?t4v2?$N$$$m$$$m$JITJQNL$N7W;;$,HyJ,:nMQAG$N7W;;$K5"Ce$9$k(B.
@code{sm1} $B$K$D$$$F$NJ8=q$O(B @code{OpenXM/doc/kan96xx} $B$K$"$k(B.
$B$J$*(B, sm1 server windows $BHG$O%P%$%J%jG[I[$7$F$$$J$$(B.
cygwin $B4D6-$G%=!<%9%3!<%I$+$i%3%s%Q%$%k$7(B, OpenXM/misc/packages/Windows
$B$K=>$$JQ99$r2C$($k$H(B sm1 $B%5!<%P$O(Bwindows $B$G$bF0:n$9$k(B.
$B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,
$BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.
$B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.
@tex
$B6u4V(B
$X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$
$B$N%I%i!<%`%3%[%b%m%872C#$N<!85$r7W;;$7$F$_$h$&(B.
$X$ $B$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
$B$^$o$kFs$D$N%k!<%W$,(B1$B<!85$N%[%b%m%8!<72$N6u4V$r$O$k(B.
$B$7$?$,$C$F(B, 1$B<!85%I%i!<%`%3%[%b%m%872$N<!85$O(B $2$ $B$G$"$k(B.
@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
$B<!85$rEz$($k(B.
@end tex
*/
/*&en
@chapter SM1 Functions
This chapter describes interface functions for
sm1 ox server @code{ox_sm1_forAsir}.
These interface functions are defined in the file @file{sm1.rr}.
The file @file{sm1.rr} is @*
at @file{$(OpenXM_HOME)/lib/asir/contrib-packages}.
The system @code{sm1} is a system to compute in the ring of differential
operators.
Many constructions of invariants
in the computational algebraic geometry reduce
to constructions in the ring of differential operators.
Documents on @code{sm1} are in
the directory @code{OpenXM/doc/kan96xx}.
The sm1 server for windows is not distributed in the binary form.
If you need to run it, compile it under the cygwin environment
following the Makefile in OpenXM/misc/packages/Windows.
All the coefficients of input polynomials should be
integers for most functions in this section.
Other functions accept rational numbers as inputs
and it will be explicitely noted in each explanation
of these functions.
@tex
Let us evaluate the dimensions of the de Rham cohomology groups
of
$X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$.
The space $X$ is a two punctured plane, so two loops that encircles the
points $x=0$ and $x=1$ respectively spans the first homology group.
Hence, the dimension of the first de Rham cohomology group is $2$.
@code{sm1} answers the dimensions of the 0th and the first
cohomology groups.
@end tex
*/
/*&C
@example
[283] sm1.deRham([x*(x-1),[x]]);
[1,2]
@end example
*/
/*&C
@noindent
The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*
The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
Grobner Deformations of Hypergeometric Differential Equations,
1999, Springer.
@url{http://www.math.kobe-u.ac.jp/KAN}
*/
/*&C
@menu
* ox_sm1_forAsir::
* sm1.start::
* sm1.sm1::
* sm1.push_int0::
* sm1.gb::
* sm1.deRham::
* sm1.hilbert::
* sm1.genericAnn::
* sm1.wTensor0::
* sm1.reduction::
* sm1.xml_tree_to_prefix_string::
* sm1.syz::
* sm1.mul::
* sm1.distraction::
* sm1.gkz::
* sm1.mgkz::
* sm1.appell1::
* sm1.appell4::
* sm1.rank::
* sm1.auto_reduce::
* sm1.slope::
* sm1.ahg::
* sm1.bfunction::
* sm1.generalized_bfunction::
* sm1.restriction::
* sm1.saturation::
@end menu
*/
/*&ja
@section @code{ox_sm1_forAsir} $B%5!<%P(B
*/
/*&en
@section @code{ox_sm1_forAsir} Server
*/
/*&en
@node ox_sm1_forAsir,,, SM1 Functions
@subsection @code{ox_sm1_forAsir}
@findex ox_sm1_forAsir
@table @t
@item ox_sm1_forAsir
:: @code{sm1} server for @code{asir}.
@end table
@itemize @bullet
@item
@code{ox_sm1_forAsir} is the @code{sm1} server started from asir
by the command @code{sm1.start}.
In the standard setting, @*
@code{ox_sm1_forAsir} =
@file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
+
@file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
+
@file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
The macro files @file{callsm1.sm1} and @file{callsm1b.sm1}
are searched from
current directory, @code{$(LOAD_SM1_PATH)},
@code{$(OpenXM_HOME)/lib/sm1},
@code{/usr/local/lib/sm1}
in this order.
@item Note for programmers: See the files
@file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
@file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
to build your own server by reading @code{sm1} macros.
@end itemize
*/
/*&ja
@node ox_sm1_forAsir,,, SM1 Functions
@subsection @code{ox_sm1_forAsir}
@findex ox_sm1_forAsir
@table @t
@item ox_sm1_forAsir
:: @code{asir} $B$N$?$a$N(B @code{sm1} $B%5!<%P(B.
@end table
@itemize @bullet
@item
$B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B
@code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.
$BI8=`E*@_Dj$G$O(B, @*
@code{ox_sm1_forAsir} =
@file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
+
@file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
+
@file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
$B$G$"$j(B, $B$3$l$i$N%^%/%m%U%!%$%k$O(B, $B0lHL$K$O(B
current directory, @code{$(LOAD_SM1_PATH)},
@code{$(OpenXM_HOME)/lib/sm1},
@code{/usr/local/lib/sm1}
$B$N=gHV$G$5$,$5$l$k(B.
@item $B%W%m%0%i%^!<$N$?$a$N%N!<%H(B:
@code{sm1} $B%^%/%m$rFI$_9~$s$G<+J,FH<+$N%5!<%P$r:n$k$K$O(B
$B<!$N%U%!%$%k$b8+$h(B
@file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
@file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
@end itemize
*/
/*&ja
@section $BH!?t0lMw(B
*/
/*&en
@section Functions
*/
/*&en
@c sort-sm1.start
@node sm1.start,,, SM1 Functions
@subsection @code{sm1.start}
@findex sm1.start
@table @t
@item sm1.start()
:: Start @code{ox_sm1_forAsir} on the localhost.
@end table
@table @var
@item return
Integer
@end table
@itemize @bullet
@item Start @code{ox_sm1_forAsir} on the localhost.
It returns the descriptor of @code{ox_sm1_forAsir}.
@item Set @code{Xm_noX = 1} to start @code{ox_sm1_forAsir}
without a debug window.
@item You might have to set suitable orders of variable by the command
@code{ord}. For example,
when you are working in the
ring of differential operators on the variable @code{x} and @code{dx}
(@code{dx} stands for
@tex $\partial/\partial x$
@end tex
),
@code{sm1} server assumes that
the variable @code{dx} is collected to the right and the variable
@code{x} is collected to the left in the printed expression.
In the example below, you must not use the variable @code{cc}
for computation in @code{sm1}.
@item The variables from @code{a} to @code{z} except @code{d} and @code{o}
and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
@code{z0}, ..., @code{z20} can be used as variables for ring of
differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).
@item The descriptor is stored in @code{static Sm1_proc}.
The descriptor can be obtained by the function
@code{sm1.get_Sm1_proc()}.
@end itemize
*/
/*&ja
@c sort-sm1.start
@node sm1.start,,, SM1 Functions
@subsection @code{sm1.start}
@findex sm1.start
@table @t
@item sm1.start()
:: localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
@end table
@table @var
@item return
$B@0?t(B
@end table
@itemize @bullet
@item localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
$B%5!<%P(B @code{ox_sm1_forAsir} $B$N<1JLHV9f$rLa$9(B.
@item @code{Xm_noX = 1} $B$H$*$/$H%5!<%P(B @code{ox_sm1_forAsir} $B$r%G%P%C%0MQ$N(B
$B%&%#%s%I%&$J$7$K5/F0$G$-$k(B.
@item $B%3%^%s%I(B @code{ord} $B$rMQ$$$FJQ?t=g=x$r@5$7$/@_Dj$7$F$*$/I,MW$,(B
$B$"$k(B.
$B$?$H$($P(B,
$BJQ?t(B @code{x} $B$H(B @code{dx} $B>e$NHyJ,:nMQAG4D(B
(@code{dx} $B$O(B
@tex $\partial/\partial x$
@end tex
$B$KBP1~(B)
$B$G7W;;$7$F$$$k$H$-(B,
@code{sm1} $B%5!<%P$O<0$r0u:~$7$?$H$-(B,
$BJQ?t(B @code{dx} $B$O1&B&$K=8$a$lJQ?t(B
@code{x} $B$O:8B&$K$"$D$a$i$l$F$$$k$H2>Dj$7$F$$$k(B.
$B<!$NNc$G$O(B, $BJQ?t(B @code{cc} $B$r(B @code{sm1} $B$G$N7W;;$N$?$a$KMQ$$$F$O$$$1$J$$(B.
@item @code{a} $B$h$j(B @code{z} $B$N$J$+$G(B, @code{d} $B$H(B @code{o} $B$r=|$$$?$b$N(B,
$B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
@code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B
$B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).
@item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.
$B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.
@end itemize
*/
/*&C
@example
[260] ord([da,a,db,b]);
[da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
......... omit ..................
]
[261] a*da;
a*da
[262] cc*dcc;
dcc*cc
[263] sm1.mul(da,a,[a]);
a*da+1
[264] sm1.mul(a,da,[a]);
a*da
@end example
*/
/*&en
@table @t
@item Reference
@code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
@code{ord}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
@code{ord}
@end table
*/
/*&en
@c sort-sm1
@node sm1.sm1,,, SM1 Functions
@subsection @code{sm1.sm1}
@findex sm1.sm1
@table @t
@item sm1.sm1(@var{p},@var{s})
:: ask the @code{sm1} server to execute the command string @var{s}.
@end table
@table @var
@item return
Void
@item p
Number
@item s
String
@end table
@itemize @bullet
@item It asks the @code{sm1} server of the descriptor number @var{p}
to execute the command string @var{s}.
(In the next example, the descriptor number is 0.)
@end itemize
*/
/*&ja
@node sm1.sm1,,, SM1 Functions
@subsection @code{sm1.sm1}
@findex sm1.sm1
@table @t
@item sm1.sm1(@var{p},@var{s})
:: $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.
@end table
@table @var
@item return
$B$J$7(B
@item p
$B?t(B
@item s
$BJ8;zNs(B
@end table
@itemize @bullet
@item $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B
$B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.
($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
@end itemize
*/
/*&C
@example
[261] sm1.sm1(0," ( (x-1)^2 ) . ");
0
[262] ox_pop_string(0);
x^2-2*x+1
[263] sm1.sm1(0," [(x*(x-1)) [(x)]] deRham ");
0
[264] ox_pop_string(0);
[1 , 2]
@end example
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
@end table
*/
/*&en
@table @t
@item Reference
@code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
@end table
*/
/*&en
@c sort-sm1.push_int0
@node sm1.push_int0,,, SM1 Functions
@subsection @code{sm1.push_int0}
@findex sm1.push_int0
@table @t
@item sm1.push_int0(@var{p},@var{f})
:: push the object @var{f} to the server with the descriptor number @var{p}.
@end table
@table @var
@item return
Void
@item p
Number
@item f
Object
@end table
@itemize @bullet
@item When @code{type(@var{f})} is 2 (recursive polynomial),
@var{f} is converted to a string (type == 7)
and is sent to the server by @code{ox_push_cmo}.
@item When @code{type(@var{f})} is 0 (zero),
it is translated to the 32 bit integer zero
on the server.
Note that @code{ox_push_cmo(@var{p},0)} sends @code{CMO_NULL} to the server.
In other words, the server does not get the 32 bit integer 0 nor
the bignum 0.
@item @code{sm1} integers are classfied into the 32 bit integer and
the bignum.
When @code{type(@var{f})} is 1 (number), it is translated to the
32 bit integer on the server.
Note that @code{ox_push_cmo(@var{p},1234)} send the bignum 1234 to the
@code{sm1} server.
@item In other cases, @code{ox_push_cmo} is called without data conversion.
@end itemize
*/
/*&ja
@c sort-sm1.push_int0
@node sm1.push_int0,,, SM1 Functions
@subsection @code{sm1.push_int0}
@findex sm1.push_int0
@table @t
@item sm1.push_int0(@var{p},@var{f})
:: $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.
@end table
@table @var
@item return
$B$J$7(B
@item p
$B?t(B
@item f
$B%*%V%8%'%/%H(B
@end table
@itemize @bullet
@item @code{type(@var{f})} $B$,(B 2 ($B:F5"B?9`<0(B) $B$N$H$-(B,
@var{f} $B$OJ8;zNs(B (type == 7) $B$KJQ49$5$l$F(B,
@code{ox_push_cmo} $B$rMQ$$$F%5!<%P$XAw$i$l$k(B.
@item @code{type(@var{f})} $B$,(B 0 (zero) $B$N$H$-$O(B,
$B%5!<%P>e$G$O(B, 32 bit $B@0?t$H2r<a$5$l$k(B.
$B$J$*(B @code{ox_push_cmo(P,0)} $B$O%5!<%P$KBP$7$F(B @code{CMO_NULL}
$B$r$*$/$k$N$G(B, $B%5!<%PB&$G$O(B, 32 bit $B@0?t$r<u$1<h$k$o$1$G$O$J$$(B.
@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.
@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
$B$*$/$k(B.
@code{ox_push_cmo(@var{p},1234)} $B$O(B bignum $B$N(B 1234 $B$r(B
@code{sm1} $B%5!<%P$K$*$/$k$3$H$KCm0U$7$h$&(B.
@item $B$=$NB>$N>l9g$K$O(B @code{ox_push_cmo} $B$r%G!<%?7?$NJQ49$J$7$K8F$S=P$9(B.
@end itemize
*/
/*&C
@example
[219] P=sm1.start();
0
[220] sm1.push_int0(P,x*dx+1);
0
[221] A=ox_pop_cmo(P);
x*dx+1
[223] type(A);
7 (string)
@end example
@example
[271] sm1.push_int0(0,[x*(x-1),[x]]);
0
[272] ox_execute_string(0," deRham ");
0
[273] ox_pop_cmo(0);
[1,2]
@end example
*/
/*&en
@table @t
@item Reference
@code{ox_push_cmo}
@end table
*/
/*&ja
@table @t
@item Reference
@code{ox_push_cmo}
@end table
*/
/*&en
@c sort-sm1.gb
@node sm1.gb,,, SM1 Functions
@subsection @code{sm1.gb}
@findex sm1.gb
@findex sm1.gb_d
@table @t
@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
:: computes the Grobner basis of @var{f} in the ring of differential
operators with the variable @var{v}.
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
:: computes the Grobner basis of @var{f} in the ring of differential
operators with the variable @var{v}.
The result will be returned as a list of distributed polynomials.
@end table
@table @var
@item return
List
@item p, q, r
Number
@item f, v, w
List
@end table
@itemize @bullet
@item
It returns the Grobner basis of the set of polynomials @var{f}
in the ring of deferential operators with the variables @var{v}.
@item
The weight vectors are given by @var{w}, which can be omitted.
If @var{w} is not given,
the graded reverse lexicographic order will be used to compute Grobner basis.
@item
The return value of @code{sm1.gb}
is the list of the Grobner basis of @var{f} and the initial
terms (when @var{w} is not given) or initial ideal (when @var{w} is given).
@item
@code{sm1.gb_d} returns the results by a list of distributed polynomials.
Monomials in each distributed polynomial are ordered in the given order.
The return value consists of
[variable names, order matrix, grobner basis in districuted polynomials,
initial monomials or initial polynomials].
@item
When a non-term order is given, the Grobner basis is computed in
the homogenized Weyl algebra (See Section 1.2 of the book of SST).
The homogenization variable h is automatically added.
@item
When the optional variable @var{q} is set, @code{sm1.gb} returns,
as the third return value, a list of
the Grobner basis and the initial ideal
with sums of monomials sorted by the given order.
Each polynomial is expressed as a string temporally for now.
When the optional variable @var{r} is set to one,
the polynomials are dehomogenized (,i.e., h is set to 1).
@end itemize
*/
/*&ja
@c sort-sm1.gb
@node sm1.gb,,, SM1 Functions
@subsection @code{sm1.gb}
@findex sm1.gb
@findex sm1.gb_d
@table @t
@item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
@item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$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.
@end table
@table @var
@item return
$B%j%9%H(B
@item p, q, r
$B?t(B
@item f, v, w
$B%j%9%H(B
@end table
@itemize @bullet
@item
@var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
@item
Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
$B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
$B%V%l%V%J4pDl$r7W;;$9$k(B.
@item
@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
( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
$B$N%j%9%H$G$"$k(B.
@item
@code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
$BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.
$BLa$jCM$O(B
[$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]
$B$G$"$k(B.
@item
Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
$BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
@item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
$BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
$B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
$B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
$BLa$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).
@end itemize
*/
/*&C
@example
[293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
[[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
@end example
*/
/*&en
In the example above,
@tex the set $\{ x \partial_x + y \partial_y -1,
y^2 \partial_y^2+2\}$
is the Gr\"obner basis of the input with respect to the
graded reverse lexicographic order such that
$ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
(the initial monominals) of the Gr\"obner basis.
@end tex
*/
/*&ja
$B>e$NNc$K$*$$$F(B,
@tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
y^2 \partial_y^2+2\}$
$B$O(B
$ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
$B$G$"$k$h$&$J(B
graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
$B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
$BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
@end tex
*/
/*&C
@example
[294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
[[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
@end example
*/
/*&en
In the example above, two monomials
@tex
$m = x^a y^b \partial_x^c \partial_y^d$ and
$m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
are firstly compared by the weight vector
{\tt (dx,dy,x,y) = (50,2,1,0)}
(i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
and when the comparison is tie, then these are
compared by the reverse lexicographic order
(i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
@end tex
*/
/*&ja
$B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
@tex
$m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
$m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
$B$O:G=i$K(B weight vector
{\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
$m'$ $B$h$jBg$-$$(B )
$B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
$B$5$l$k(B).
@end tex
*/
/*&C
@example
[294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
map(print,F[2][0])$
map(print,F[2][1])$
@end example
*/
/*&C
@example
[595]
sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
[x,y],[[dx,1,x,-1],[dy,1]]]);
[[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],
[x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
[596]
sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
"x,y",[[dx,1,x,-1],[dy,1]]]);
[[[e0,x,y,H,E,dx,dy,h],
[[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
[0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
[0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
[0,0,0,0,0,0,0,1]]],
[[(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)*
<<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
,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
,1,3>>],
[(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)*<
<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
,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
@end example
*/
/*&en
@table @t
@item Reference
@code{sm1.reduction}, @code{sm1.rat_to_p}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm1.reduction}, @code{sm1.rat_to_p}
@end table
*/
/*&en
@c sort-sm1.deRham
@node sm1.deRham,,, SM1 Functions
@subsection @code{sm1.deRham}
@findex sm1.deRham
@table @t
@item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
:: ask the server to evaluate the dimensions of the de Rham cohomology groups
of C^n - (the zero set of @var{f}=0).
@end table
@table @var
@item return
List
@item p
Number
@item f
String or polynomial
@item v
List
@end table
@itemize @bullet
@item It returns the dimensions of the de Rham cohomology groups
of X = C^n \ V(@var{f}).
In other words, it returns
[dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
@item @var{v} is a list of variables. n = @code{length(@var{v})}.
@item
@code{sm1.deRham} requires huge computer resources.
For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
is already very hard.
@item
To efficiently analyze the roots of b-function, @code{ox_asir} should be used
from @code{ox_sm1_forAsir}.
It is recommended to load the communication module for @code{ox_asir}
by the command @*
@code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
This command is automatically executed when @code{ox_sm1_forAsir} is started.
@item If you make an interruption to the function @code{sm1.deRham}
by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
mode. So, it is strongly recommended to execute the command
@code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
@end itemize
*/
/*&ja
@c sort-sm1.deRham
@node sm1.deRham,,, SM1 Functions
@subsection @code{sm1.deRham}
@findex sm1.deRham
@table @t
@item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
:: $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.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item f
$BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
@item v
$B%j%9%H(B
@end table
@itemize @bullet
@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.
$B$9$J$o$A(B,
[dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
$B$rLa$9(B.
@item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
@item
@code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
$B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
$B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
@item
b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
$B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
@code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
$B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
$B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
@item
@code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
$B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
$B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
$B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
@end itemize
*/
/*&C
@example
[332] sm1.deRham([x^3-y^2,[x,y]]);
[1,1,0]
[333] sm1.deRham([x*(x-1),[x]]);
[1,2]
@end example
*/
/*&en
@table @t
@item Reference
@code{sm1.start}, @code{deRham} (sm1 command)
@item Algorithm:
Oaku, Takayama, An algorithm for de Rham cohomology groups of the
complement of an affine variety via D-module computation,
Journal of pure and applied algebra 139 (1999), 201--233.
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm1.start}, @code{deRham} (sm1 command)
@item Algorithm:
Oaku, Takayama, An algorithm for de Rham cohomology groups of the
complement of an affine variety via D-module computation,
Journal of pure and applied algebra 139 (1999), 201--233.
@end table
*/
/*&en
@c sort-sm1.hilbert
@node sm1.hilbert,,, SM1 Functions
@subsection @code{sm1.hilbert}
@findex sm1.hilbert
@findex hilbert_polynomial
@table @t
@item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
:: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
@item hilbert_polynomial(@var{f},@var{v})
:: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
@end table
@table @var
@item return
Polynomial
@item p
Number
@item f, v
List
@end table
@itemize @bullet
@item It returns the Hilbert polynomial h(k) of the set of polynomials
@var{f}
with respect to the set of variables @var{v}.
@item
h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
degree is less than or equal to k and I is the ideal generated by the
set of polynomials @var{f}.
@item
Note for sm1.hilbert:
For an efficient computation, it is preferable that
the set of polynomials @var{f} is a set of monomials.
In fact, this function firstly compute a Grobner basis of @var{f}, and then
compute the Hilbert polynomial of the initial monomials of the basis.
If the input @var{f} is already a Grobner
basis, a Grobner basis is recomputed in this function,
which is a waste of time and Grobner basis computation in the ring of
polynomials in @code{sm1} is slower than in @code{asir}.
@end itemize
*/
/*&ja
@c sort-sm1.hilbert
@node sm1.hilbert,,, SM1 Functions
@subsection @code{sm1.hilbert}
@findex sm1.hilbert
@findex hilbert_polynomial
@table @t
@item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
:: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
@item hilbert_polynomial(@var{f},@var{v})
:: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
@end table
@table @var
@item return
$BB?9`<0(B
@item p
$B?t(B
@item f, v
$B%j%9%H(B
@end table
@itemize @bullet
@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)
$B$r7W;;$9$k(B.
@item
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
$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.
@item
sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
$B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
$B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
$B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
$B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
$BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
@end itemize
*/
/*&C
@example
[346] load("katsura")$
[351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
32
@end example
@example
[279] load("katsura")$
[280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
[281] dp_ord();
0
[282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
[(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>>,
(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>>,
(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>>,
(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>>,
(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>>,
(1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
[283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
[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,
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,
u5^4*u2,u5^4*u1,u5^6]
[284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
32
@end example
*/
/*&en
@table @t
@item Reference
@code{sm1.start}, @code{sm1.gb}, @code{longname}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm1.start}, @code{sm1.gb}, @code{longname}
@end table
*/
/*&en
@c sort-sm1.genericAnn
@node sm1.genericAnn,,, SM1 Functions
@subsection @code{sm1.genericAnn}
@findex sm1.genericAnn
@table @t
@item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
:: It computes the annihilating ideal for @var{f}^s.
@var{v} is the list of variables. Here, s is @var{v}[0] and
@var{f} is a polynomial in the variables @code{rest}(@var{v}).
@end table
@table @var
@item return
List
@item p
Number
@item f
Polynomial
@item v
List
@end table
@itemize @bullet
@item This function computes the annihilating ideal for @var{f}^s.
@var{v} is the list of variables. Here, s is @var{v}[0] and
@var{f} is a polynomial in the variables @code{rest}(@var{v}).
@end itemize
*/
/*&ja
@c sort-sm1.genericAnn
@node sm1.genericAnn,,, SM1 Functions
@subsection @code{sm1.genericAnn}
@findex sm1.genericAnn
@table @t
@item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
:: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
@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,
@var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item f
$BB?9`<0(B
@item v
$B%j%9%H(B
@end table
@itemize @bullet
@item $B$3$NH!?t$O(B,
@var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
@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,
@var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
@end itemize
*/
/*&C
@example
[595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
[-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]
@end example
*/
/*&en
@table @t
@item Reference
@code{sm1.start}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm1.start}
@end table
*/
/*&en
@c sort-sm1.wTensor0
@node sm1.wTensor0,,, SM1 Functions
@subsection @code{sm1.wTensor0}
@findex sm1.wTensor0
@table @t
@item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
:: It computes the D-module theoretic 0-th tensor product
of @var{f} and @var{g}.
@end table
@table @var
@item return
List
@item p
Number
@item f, g, v, w
List
@end table
@itemize @bullet
@item
It returns the D-module theoretic 0-th tensor product
of @var{f} and @var{g}.
@item
@var{v} is a list of variables.
@var{w} is a list of weights. The integer @var{w}[i] is
the weight of the variable @var{v}[i].
@item
@code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
which requires a generic weight
vector @var{w} to compute the restriction.
If @var{w} is not generic, the computation fails.
@item Let F and G be solutions of @var{f} and @var{g} respectively.
Intuitively speaking, the 0-th tensor product is a system of
differential equations which annihilates the function FG.
@item The answer is a submodule of a free module D^r in general even if
the inputs @var{f} and @var{g} are left ideals of D.
@end itemize
*/
/*&ja
@c sort-sm1.wTensor0
@node sm1.wTensor0,,, SM1 Functions
@subsection @code{sm1.wTensor0}
@findex sm1.wTensor0
@table @t
@item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
:: @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
$B7W;;$9$k(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item f, g, v, w
$B%j%9%H(B
@end table
@itemize @bullet
@item
@var{f} $B$H(B @var{g} $B$N(B
D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
@item
@var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
@var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
$B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
@item
@code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
$B$r$h$s$G$$$k(B.
@code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
$B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
@item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
$BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
@item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
$B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
@end itemize
*/
/*&C
@example
[258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
[[-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],
[-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],
[y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
@end example
*/
/*&en
@c sort-sm1.reduction
@node sm1.reduction,,, SM1 Functions
@subsection @code{sm1.reduction}
@findex sm1.reduction
@table @t
@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
::
@end table
@table @var
@item return
List
@item f
Polynomial
@item g, v, w
List
@item p
Number (the process number of ox_sm1)
@end table
@itemize @bullet
@item It reduces @var{f} by the set of polynomial @var{g}
in the homogenized Weyl algebra; it applies the
division algorithm to @var{f}. The set of variables is @var{v} and
@var{w} is weight vectors to determine the order, which can be ommited.
@code{sm1.reduction_noH} is for the Weyl algebra.
@item The return value is of the form
[r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
c0 f + c1 g1 + ... + cm gm = r.
r/c0 is the normal form.
@item The function reduction reduces reducible terms that appear
in lower order terms.
@item The functions
sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
are for distributed polynomials.
@end itemize
*/
/*&ja
@node sm1.reduction,,, SM1 Functions
@subsection @code{sm1.reduction}
@findex sm1.reduction
@table @t
@item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
::
@end table
@table @var
@item return
$B%j%9%H(B
@item f
$BB?9`<0(B
@item g, v, w
$B%j%9%H(B
@item p
$B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
@end table
@itemize @bullet
@item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
$BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
$B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
$BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
@var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
$B>JN,$7$F$b$h$$(B.
@code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
@item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
[r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
c0 f + c1 g1 + ... + cm gm = r
$B$,$J$j$?$D(B.
r/c0 $B$,(B normal form $B$G$"$k(B.
@item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
@item $BH!?t(B
sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
$B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
@end itemize
*/
/*&C
@example
[259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
[x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
[260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
[0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
@end example
*/
/*&en
@table @t
@item Reference
@code{sm1.start}, @code{d_true_nf}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm1.start}, @code{d_true_nf}
@end table
*/
/*&en
@node sm1.xml_tree_to_prefix_string,,, SM1 Functions
@subsection @code{sm1.xml_tree_to_prefix_string}
@findex sm1.xml_tree_to_prefix_string
@table @t
@item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
:: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
@end table
@table @var
@item return
String
@item p
Number
@item s
String
@end table
@itemize @bullet
@item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
@item This function should be moved to om_* in a future.
@item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
asir has not yet understood this CMO.
@item @code{java} execution environment is required.
(For example, @code{/usr/local/jdk1.1.8/bin} should be in the
command search path.)
@end itemize
*/
/*&ja
@node sm1.xml_tree_to_prefix_string,,, SM1 Functions
@subsection @code{sm1.xml_tree_to_prefix_string}
@findex sm1.xml_tree_to_prefix_string
@table @t
@item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
:: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
@end table
@table @var
@item return
String
@item p
Number
@item s
String
@end table
@itemize @bullet
@item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
@item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
@item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
$B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
@item @code{java} $B$N<B9T4D6-$,I,MW(B.
($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.)
@end itemize
*/
/*&C
@example
[263] load("om");
1
[270] F=om_xml(x^4-1);
control: wait OX
Trying to connect to the server... Done.
<OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
<OMS name="times" cd="basic"/><OMA>
<OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
<OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
<OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
<OMI>-1</OMI></OMA></OMA></OMOBJ>
[271] sm1.xml_tree_to_prefix_string(F);
basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
@end example
*/
/*&en
@table @t
@item Reference
@code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
@end table
*/
/*&en
@c sort-sm1.syz
@node sm1.syz,,, SM1 Functions
@subsection @code{sm1.syz}
@findex sm1.syz
@findex sm1.syz_d
@table @t
@item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
:: computes the syzygy of @var{f} in the ring of differential
operators with the variable @var{v}.
@end table
@table @var
@item return
List
@item p
Number
@item f, v, w
List
@end table
@itemize @bullet
@item
The return values is of the form
[@var{s},[@var{g}, @var{m}, @var{t}]].
Here @var{s} is the syzygy of @var{f} in the ring of differential
operators with the variable @var{v}.
@var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
basis @var{g}.
@var{t} is the syzygy of the Gr\"obner basis @var{g}.
In summary, @var{g} = @var{m} @var{f} and
@var{s} @var{f} = 0 hold as matrices.
@item
The weight vectors are given by @var{w}, which can be omitted.
If @var{w} is not given,
the graded reverse lexicographic order will be used to compute Grobner basis.
@item
When a non-term order is given, the Grobner basis is computed in
the homogenized Weyl algebra (See Section 1.2 of the book of SST).
The homogenization variable h is automatically added.
@end itemize
*/
/*&ja
@c sort-sm1.syz
@node sm1.syz,,, SM1 Functions
@subsection @code{sm1.syz}
@findex sm1.syz
@findex sm1.syz_d
@table @t
@item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
:: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item f, v, w
$B%j%9%H(B
@end table
@itemize @bullet
@item
$BLa$jCM$O<!$N7A$r$7$F$$$k(B:
[@var{s},[@var{g}, @var{m}, @var{t}]].
$B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
syzygy $B$G$"$k(B.
@var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
@var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
@var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
@var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
$B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
@var{g} = @var{m} @var{f} ,
@var{s} @var{f} = 0.
@item
Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
$B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
$B%V%l%V%J4pDl$r7W;;$9$k(B.
@item
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).
$BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
@end itemize
*/
/*&C
@example
[293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
[[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
[[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
[[1,0],[y*dy,-1]], transformation matrix
[[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
@end example
*/
/*&C
@example
[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]]]);
[[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
[[[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
[h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
[[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
[[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]]]]
@end example
*/
/*&en
@node sm1.mul,,, SM1 Functions
@subsection @code{sm1.mul}
@findex sm1.mul
@table @t
@item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
:: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
@end table
@table @var
@item return
Polynomial or List
@item p
Number
@item f, g
Polynomial or List
@item v
List
@end table
@itemize @bullet
@item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
@item @code{sm1.mul_h} is for homogenized Weyl algebra.
@item BUG: @code{sm1.mul(p0*dp0,1,[p0])} returns
@code{dp0*p0+1}.
A variable order such that d-variables come after non-d-variables
is necessary for the correct computation.
@end itemize
*/
/*&ja
@node sm1.mul,,, SM1 Functions
@subsection @code{sm1.mul}
@findex sm1.mul
@table @t
@item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
:: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
$B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
@end table
@table @var
@item return
$BB?9`<0$^$?$O%j%9%H(B
@item p
$B?t(B
@item f, g
$BB?9`<0$^$?$O%j%9%H(B
@item v
$B%j%9%H(B
@end table
@itemize @bullet
@item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
$B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
@item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
@item BUG: @code{sm1.mul(p0*dp0,1,[p0])} $B$O(B
@code{dp0*p0+1} $B$rLa$9(B.
d$BJQ?t$,8e$m$K$/$k$h$&$JJQ?t=g=x$,$O$$$C$F$$$J$$$H(B, $B$3$N4X?t$O@5$7$$Ez$($rLa$5$J$$(B.
@end itemize
*/
/*&C
@example
[277] sm1.mul(dx,x,[x]);
x*dx+1
[278] sm1.mul([x,y],[1,2],[x,y]);
x+2*y
[279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
[[x+2,y+4],[3*x+4,3*y+8]]
@end example
*/
/*&en
@node sm1.distraction,,, SM1 Functions
@subsection @code{sm1.distraction}
@findex sm1.distraction
@table @t
@item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
:: ask the @code{sm1} server to compute the distraction of @var{f}.
@end table
@table @var
@item return
List
@item p
Number
@item f
Polynomial
@item v,x,d,s
List
@end table
@itemize @bullet
@item It asks the @code{sm1} server of the descriptor number @var{p}
to compute the distraction of @var{f} in the ring of differential
operators with variables @var{v}.
@item @var{x} is a list of x-variables and @var{d} is that of d-variables
to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
@item Distraction is roughly speaking to replace x*dx by a single variable x.
See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
@end itemize
*/
/*&ja
@node sm1.distraction,,, SM1 Functions
@subsection @code{sm1.distraction}
@findex sm1.distraction
@table @t
@item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
:: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item f
$BB?9`<0(B
@item v,x,d,s
$B%j%9%H(B
@end table
@itemize @bullet
@item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
@var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
@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
$B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
@item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
$B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
@end itemize
*/
/*&C
@example
[280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
x
[281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
x^2-x
[282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
x^2+3*x+2
[283] fctr(@@);
[[1,1],[x+1,1],[x+2,1]]
[284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
(x^2-x)*dy+x*y
@end example
*/
/*&en
@table @t
@item Reference
@code{distraction2(sm1)},
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{distraction2(sm1)},
@end table
*/
/*&en
@node sm1.gkz,,, SM1 Functions
@subsection @code{sm1.gkz}
@findex sm1.gkz
@table @t
@item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
:: Returns the GKZ system (A-hypergeometric system) associated to the matrix
@var{A} with the parameter vector @var{B}.
@end table
@table @var
@item return
List
@item p
Number
@item A, B
List
@end table
@itemize @bullet
@item Returns the GKZ hypergeometric system
(A-hypergeometric system) associated to the matrix
@end itemize
*/
/*&ja
@node sm1.gkz,,, SM1 Functions
@subsection @code{sm1.gkz}
@findex sm1.gkz
@table @t
@item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
:: $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.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item A, B
$B%j%9%H(B
@end table
@itemize @bullet
@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.
@end itemize
*/
/*&C
@example
[280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
[[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
-dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
[x1,x2,x3,x4]]
@end example
*/
/*&en
@node sm1.mgkz,,, SM1 Functions
@subsection @code{sm1.mgkz}
@findex sm1.mgkz
@table @t
@item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
:: Returns the modified GKZ system (A-hypergeometric system) associated to the matrix
@var{A} and the weight @var{w} with the parameter vector @var{B}.
@end table
@table @var
@item return
List
@item p
Number
@item A, W, B
List
@end table
@itemize @bullet
@item Returns the modified GKZ hypergeometric system
(A-hypergeometric system) associated to the matrix
@item http://arxiv.org/abs/0707.0043
@end itemize
*/
/*&ja
@node sm1.mgkz,,, SM1 Functions
@subsection @code{sm1.mgkz}
@findex sm1.mgkz
@table @t
@item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
:: $B9TNs(B @var{A}, weight @var{W} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B modified GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item A, W, B
$B%j%9%H(B
@end table
@itemize @bullet
@item $B9TNs(B @var{A}, weight vector @var{W} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B modified GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
@item http://arxiv.org/abs/0707.0043
@end itemize
*/
/*&C
@example
[280] sm1.mgkz([ [[1,2,3]], [1,2,1], [a/2]]);
[[6*x3*dx3+4*x2*dx2+2*x1*dx1-a,-x4*dx4+x3*dx3+2*x2*dx2+x1*dx1,
-dx2+dx1^2,-x4^2*dx3+dx1*dx2],[x1,x2,x3,x4]]
Modified A-hypergeometric system for
A=(1,2,3), w=(1,2,1), beta=(a/2).
@end example
*/
/*&en
@node sm1.appell1,,, SM1 Functions
@subsection @code{sm1.appell1}
@findex sm1.appell1
@table @t
@item sm1.appell1(@var{a}|proc=@var{p})
:: Returns the Appell hypergeometric system F_1 or F_D.
@end table
@table @var
@item return
List
@item p
Number
@item a
List
@end table
@itemize @bullet
@item Returns the hypergeometric system for the Lauricella function
F_D(a,b1,b2,...,bn,c;x1,...,xn)
where @var{a} =(a,c,b1,...,bn).
When n=2, the Lauricella function is called the Appell function F_1.
The parameters a, c, b1, ..., bn may be rational numbers.
@item It does not call sm1 function appell1. As a concequence,
when parameters are rational or symbolic, this function also works
as well as integral parameters.
@end itemize
*/
/*&ja
@node sm1.appell1,,, SM1 Functions
@subsection @code{sm1.appell1}
@findex sm1.appell1
@table @t
@item sm1.appell1(@var{a}|proc=@var{p})
:: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item a
$B%j%9%H(B
@end table
@itemize @bullet
@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
F_D(a,b1,b2,...,bn,c;x1,...,xn)
$B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
@var{a} =(a,c,b1,...,bn).
$B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
@item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B
$B@5$7$/F0$/(B.
@end itemize
*/
/*&C
@example
[281] sm1.appell1([1,2,3,4]);
[[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
(-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
((-x2+x1)*dx1+3)*dx2-4*dx1], equations
[x1,x2]] the list of variables
[282] sm1.gb(@@);
[[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
+(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
+(-4*x2-4*x1)*dx1-4,
(x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
+(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
[x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
[283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
3
[285] Mu=2$ Beta = 1/3$
[287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
4
@end example
*/
/*&en
@node sm1.appell4,,, SM1 Functions
@subsection @code{sm1.appell4}
@findex sm1.appell4
@table @t
@item sm1.appell4(@var{a}|proc=@var{p})
:: Returns the Appell hypergeometric system F_4 or F_C.
@end table
@table @var
@item return
List
@item p
Number
@item a
List
@end table
@itemize @bullet
@item Returns the hypergeometric system for the Lauricella function
F_4(a,b,c1,c2,...,cn;x1,...,xn)
where @var{a} =(a,b,c1,...,cn).
When n=2, the Lauricella function is called the Appell function F_4.
The parameters a, b, c1, ..., cn may be rational numbers.
@item @item It does not call sm1 function appell4. As a concequence,
when parameters are rational or symbolic, this function also works
as well as integral parameters.
@end itemize
*/
/*&ja
@node sm1.appell4,,, SM1 Functions
@subsection @code{sm1.appell4}
@findex sm1.appell4
@table @t
@item sm1.appell4(@var{a}|proc=@var{p})
:: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
@end table
@table @var
@item return
$B%j%9%H(B
@item p
$B?t(B
@item a
$B%j%9%H(B
@end table
@itemize @bullet
@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
F_C(a,b,c1,c2,...,cn;x1,...,xn)
$B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
@var{a} =(a,b,c1,...,cn).
$B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
@item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B
$B@5$7$/F0$/(B.
@end itemize
*/
/*&C
@example
[281] sm1.appell4([1,2,3,4]);
[[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
(-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
equations
[x1,x2]] the list of variables
[282] sm1.rank(@@);
4
@end example
*/
/*&en
@node sm1.rank,,, SM1 Functions
@subsection @code{sm1.rank}
@findex sm1.rank
@table @t
@item sm1.rank(@var{a}|proc=@var{p})
:: Returns the holonomic rank of the system of differential equations @var{a}.
@end table
@table @var
@item return
Number
@item p
Number
@item a
List
@end table
@itemize @bullet
@item It evaluates the dimension of the space of holomorphic solutions
at a generic point of the system of differential equations @var{a}.
The dimension is called the holonomic rank.
@item @var{a} is a list consisting of a list of differential equations and
a list of variables.
@item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
holonomic. It is generally faster than @code{sm1.rank}.
@end itemize
*/
/*&ja
@node sm1.rank,,, SM1 Functions
@subsection @code{sm1.rank}
@findex sm1.rank
@table @t
@item sm1.rank(@var{a}|proc=@var{p})
:: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
@end table
@table @var
@item return
$B?t(B
@item p
$B?t(B
@item a
$B%j%9%H(B
@end table
@itemize @bullet
@item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
$BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
@item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
@item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
$B$b(B holonomic rank $B$rLa$9(B.
$B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
@end itemize
*/
/*&C
@example
[284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
[[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
-dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
[x1,x2,x3,x4]]
[285] sm1.rrank(@@);
4
[286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
[[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
-dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
[x1,x2,x3,x4]]
[287] sm1.rrank(@@);
5
@end example
*/
/*&en
@node sm1.auto_reduce,,, SM1 Functions
@subsection @code{sm1.auto_reduce}
@findex sm1.auto_reduce
@table @t
@item sm1.auto_reduce(@var{s}|proc=@var{p})
:: Set the flag "AutoReduce" to @var{s}.
@end table
@table @var
@item return
Number
@item p
Number
@item s
Number
@end table
@itemize @bullet
@item If @var{s} is 1, then all Grobner bases to be computed
will be the reduced Grobner bases.
@item If @var{s} is 0, then Grobner bases are not necessarily the reduced
Grobner bases. This is the default.
@end itemize
*/
/*&ja
@node sm1.auto_reduce,,, SM1 Functions
@subsection @code{sm1.auto_reduce}
@findex sm1.auto_reduce
@table @t
@item sm1.auto_reduce(@var{s}|proc=@var{p})
:: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
@end table
@table @var
@item $BLa$jCM(B
$B?t(B
@item p
$B?t(B
@item s
$B?t(B
@end table
@itemize @bullet
@item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
reduced $B%0%l%V%J4pDl$H$J$k(B.
@item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
@end itemize
*/
/*&en
@node sm1.slope,,, SM1 Functions
@subsection @code{sm1.slope}
@findex sm1.slope
@table @t
@item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
:: Returns the slopes of differential equations @var{ii}.
@end table
@table @var
@item return
List
@item p
Number
@item ii
List (equations)
@item v
List (variables)
@item f_filtration
List (weight vector)
@item v_filtration
List (weight vector)
@end table
@itemize @bullet
@item @code{sm1.slope} returns the (geometric) slopes
of the system of differential equations @var{ii}
along the hyperplane specified by
the V filtration @var{v_filtration}.
@item @var{v} is a list of variables.
@item The return value is a list of lists.
The first entry of each list is the slope and the second entry
is the weight vector for which the microcharacteristic variety is
not bihomogeneous.
@end itemize
@noindent
Algorithm:
see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
Note that the signs of the slopes s' are negative, but the absolute values -s'
of the slopes are returned.
In other words, when pF+qV is the gap, -s'=q/p is returned.
Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).
The number s satisfies 1<= s.
We have r=s-1=-1/s', and kappa=1/r=-s', which is used 1/Gamma(1+m*r) factor and exp(-tau^kappa)
in the Borel and Laplace transformations respectively.
*/
/*&ja
@node sm1.slope,,, SM1 Functions
@subsection @code{sm1.slope}
@findex sm1.slope
@table @t
@item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
:: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
@end table
@table @var
@item return
$B?t(B
@item p
$B?t(B
@item ii
$B%j%9%H(B ($BJ}Dx<0(B)
@item v
$B%j%9%H(B ($BJQ?t(B)
@item f_filtration
$B%j%9%H(B (weight vector)
@item v_filtration
$B%j%9%H(B (weight vector)
@end table
@itemize @bullet
@item @code{sm1.slope} $B$O(B
$BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
$B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
@item @var{v} $B$OJQ?t$N%j%9%H(B.
@item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
$B@.J,%j%9%H$NBh(B 1 $BMWAG$,(B slope, $BBh(B 2 $BMWAG$O(B, $B$=$N(B weight vector $B$KBP1~$9$k(B
microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
@end itemize
@noindent
Algorithm:
"A.Assi, F.J.Castro-Jimenez and J.M.Granger,
How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
$B$r$_$h(B.
Slope s' $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
Slope $B$N@dBPCM(B -s' $B$rLa$9(B.
$B$D$^$j(B pF+qV $B$,(Bmicro$BFC@-B?MMBN$N(Bgap$B$G$"$k$H$-(B, -s'=q/p $B$rLa$9(B.
$B:G6a$NJ88%$G$O(B s=1-1/s' $B$r(B slope $B$H8F$s$G$$$k(B. $B2r$O(B O(s) $B$KB0$9$k(B.
$B?t(B s $B$O(B 1<= s $B$rK~$9(B.
r=s-1=-1/s' $B$*$h$S(B kappa=1/r=-s' $B$G$"$k(B.
$B$3$l$i$N?t$O(BBorel and Laplace $BJQ49$K$*$$$F$=$l$>$l(B 1/Gamma(1+m*r) factor,
exp(-tau^kappa) $B9`$H$7$F;H$o$l$k(B.
*/
/*&C
@example
[284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
[285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
[286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
(* This is an interesting example given by Laura Matusevich,
June 9, 2001 *)
[287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
@end example
*/
/*&en
@table @t
@item Reference
@code{sm.gb}
@end table
*/
/*&ja
@table @t
@item $B;2>H(B
@code{sm.gb}
@end table
*/
/*&en
@include sm1-auto.en
*/
/*&ja
@include sm1-auto.ja
*/
end$