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Annotation of OpenXM/src/asir-doc/parts/algnum.texi, Revision 1.1.1.1

1.1       noro        1: @node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top
                      2: @chapter $BBe?tE*?t$K4X$9$k1i;;(B
                      3:
                      4: @menu
                      5: * $BBe?tE*?t$NI=8=(B::
                      6: * $BBe?tE*?t$N1i;;(B::
                      7: * $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B::
                      8: * $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B::
                      9: @end menu
                     10:
                     11: @node $BBe?tE*?t$NI=8=(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
                     12: @section $BBe?tE*?t$NI=8=(B
                     13:
                     14: @noindent
                     15: @b{Asir} $B$K$*$$$F$O(B, $BBe?tBN$H$$$&BP>]$ODj5A$5$l$J$$(B.
                     16: $BFHN)$7$?BP>]$H$7$FDj5A$5$l$k$N$O(B, $BBe?tE*?t$G$"$k(B.
                     17: $BBe?tBN$O(B, $BM-M}?tBN$K(B, $BBe?tE*?t$rM-8B8D(B
                     18: $B=g<!E:2C$7$?BN$H$7$F2>A[E*$KDj5A$5$l$k(B. $B?7$?$JBe?tE*?t$O(B, $BM-M}?t$*$h$S(B
                     19: $B$3$l$^$GDj5A$5$l$?Be?tE*?t$NB?9`<0$r78?t$H$9$k(B 1 $BJQ?tB?9`<0(B
                     20: $B$rDj5AB?9`<0$H$7$FDj5A$5$l$k(B. $B0J2<(B, $B$"$kDj5AB?9`<0$N:,$H$7$F(B
                     21: $BDj5A$5$l$?Be?tE*?t$r(B, @code{root} $B$H8F$V$3$H$K$9$k(B.
                     22:
                     23: @example
                     24: [0] A0=newalg(x^2+1);
                     25: (#0)
                     26: [1] A1=newalg(x^3+A0*x+A0);
                     27: (#1)
                     28: [2]  [type(A0),ntype(A0)];
                     29: [1,2]
                     30: @end example
                     31:
                     32: @noindent
                     33: $B$3$NNc$G$O(B, @code{A0} $B$O(B @code{x^2+1=0} $B$N:,(B, @code{A1} $B$O(B, $B$=$N(B @code{A0}
                     34: $B$r78?t$K4^$`(B @code{x^3+A0*x+A0=0} $B$N:,$H$7$FDj5A$5$l$F$$$k(B.
                     35:
                     36: @noindent
                     37: @code{newalg()} $B$N0z?t$9$J$o$ADj5AB?9`<0$K$O<!$N$h$&$J@)8B$,$"$k(B.
                     38:
                     39: @enumerate
                     40: @item
                     41: $BDj5AB?9`<0$O(B 1 $BJQ?tB?9`<0$G$J$1$l$P$J$i$J$$(B.
                     42:
                     43:
                     44: @item
                     45: @code{newalg()} $B$N0z?t$G$"$kDj5AB?9`<0$O(B, $BBe?tE*?t$r4^$`<0$N4JC12=$N$?(B
                     46: $B$a$KMQ$$$i$l$k(B. $B$3$N4JC12=$O(B, $BAH$_9~$_H!?t(B @code{srem()} $B$KAjEv$9$kFb(B
                     47: $BIt%k!<%A%s$rMQ$$$F9T$o$l$k(B. $B$3$N$?$a(B, $BDj5AB?9`<0$N<g78?t$O(B, $BM-M}?t$K(B
                     48: $B$J$C$F$$$kI,MW$,$"$k(B.
                     49:
                     50: @item
                     51: $BDj5AB?9`<0$N78?t$O(B $B$9$G$KDj5A$5$l$F$$$k(B @code{root} $B$NM-M}?t78?tB?9`<0(B
                     52: $B$G$J$1$l$P$J$i$J$$(B.
                     53:
                     54: @item
                     55: $BDj5AB?9`<0$O(B, $B$=$N78?t$K4^$^$l$kA4$F$N(B @code{root} $B$rM-M}?t$KE:2C$7$?(B
                     56: $BBN>e$G4{Ls$G$J$1$l$P$J$i$J$$(B.
                     57: @end enumerate
                     58:
                     59: @noindent
                     60: @code{newalg()} $B$,9T$&0z?t%A%'%C%/$O(B, 1 $B$*$h$S(B 2 $B$N$_$G$"$k(B.
                     61: $BFC$K(B, $B0z?t$NDj5AB?9`<0$N4{Ls@-$OA4$/%A%'%C%/$5$l$J$$(B. $B$3$l$O(B
                     62: $B4{Ls@-$N%A%'%C%/$,B?Bg$J7W;;NL$rI,MW$H$9$k$?$a$G(B, $B$3$NE@$K4X$7$F$O(B,
                     63: $B%f!<%6$N@UG$$KG$$5$l$F$$$k(B.
                     64:
                     65: @noindent
                     66: $B0lC6(B @code{newalg()} $B$K$h$C$FDj5A$5$l$?Be?tE*?t$O(B, $B?t$H$7$F$N<1JL;R$r;}$A(B,
                     67: $B$^$?(B, $B?t$NCf$G$OBe?tE*?t$H$7$F$N<1JL;R$r;}$D(B. (@code{type()}, @code{vtype()}
                     68: $B;2>H(B.) $B$5$i$K(B, $BM-M}?t$H(B, @code{root} $B$NM-M}<0$bF1MM$KBe?tE*?t$H$J$k(B.
                     69:
                     70: @example
                     71: [87] N=(A0^2+A1)/(A1^2-A0-1);
                     72: ((#1+#0^2)/(#1^2-#0-1))
                     73: [88] [type(N),ntype(N)];
                     74: [1,2]
                     75: @end example
                     76:
                     77: @noindent
                     78: $BNc$+$i$o$+$k$h$&$K(B, @code{root}$B$O(B @code{#@var{n}}
                     79: $B$HI=<($5$l$k(B. $B$7$+$7(B, $B%f!<%6$O$3$N7A$G$OF~NO$G$-$J$$(B. @code{root} $B$O(B
                     80: $BJQ?t$K3JG<$7$FMQ$$$k$+(B, $B$"$k$$$O(B @code{alg(@var{n})} $B$K$h$j<h$j=P$9(B.
                     81: $B$^$?(B, $B8zN($OMn$A$k$,(B, $BA4$/F1$80z?t(B ($BJQ?t$O0[$J$C$F$$$F$b$h$$(B) $B$K$h$j(B
                     82: @code{newalg()} $B$r8F$Y$P(B, $B?7$7$$Be?tE*?t$ODj5A$5$l$:$K4{$KDj5A$5$l$?(B
                     83: $B$b$N$,F@$i$l$k(B.
                     84:
                     85: @example
                     86: [90] alg(0);
                     87: (#0)
                     88: [91] newalg(t^2+1);
                     89: (#0)
                     90: @end example
                     91:
                     92: @noindent
                     93: @code{root} $B$NDj5AB?9`<0$O(B, @code{defpoly()} $B$K$h$j<h$j=P$;$k(B.
                     94:
                     95: @example
                     96: [96] defpoly(A0);
                     97: t#0^2+1
                     98: [97] defpoly(A1);
                     99: t#1^3+t#0*t#1+t#0
                    100: @end example
                    101:
                    102: @noindent
                    103: $B$3$3$G8=$l$?(B, @code{t#0}, @code{t#1} $B$O$=$l$>$l(B @code{#0}, @code{#1} $B$K(B
                    104: $BBP1~$9$kITDj85$G$"$k(B. $B$3$l$i$b%f!<%6$,F~NO$9$k$3$H$O$G$-$J$$(B.
                    105: @code{var()} $B$G<h$j=P$9$+(B, $B$"$k$$$O(B @code{algv(@var{n})} $B$K$h$j<h$j=P$9(B.
                    106:
                    107: @example
                    108: [98] var(@@);
                    109: t#1
                    110: [99] algv(0);
                    111: t#0
                    112: [100]
                    113: @end example
                    114:
                    115: @node $BBe?tE*?t$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
                    116: @section $BBe?tE*?t$N1i;;(B
                    117:
                    118: @noindent
                    119: $BA0@a$G(B, $BBe?tE*?t$NI=8=(B, $BDj5A$K$D$$$F=R$Y$?(B. $B$3$3$G$O(B, $BBe?tE*?t$rMQ$$$?(B
                    120: $B1i;;$K$D$$$F=R$Y$k(B. $BBe?tE*?t$K4X$7$F$O(B, $BAH$_9~$_H!?t$H$7$FDs6!$5$l$F$$$k(B
                    121: $B5!G=$O$4$/>/?t$G(B, $BBgItJ,$O%f!<%6Dj5AH!?t$K$h$j<B8=$5$l$F$$$k(B. $B%U%!%$%k(B
                    122: $B$O(B, @samp{sp} $B$G(B, @samp{gr} $B$HF1MM(B @b{Asir} $B$NI8=`%i%$%V%i%j%G%#%l%/%H%j(B
                    123: $B$K$*$+$l$F$$$k(B.
                    124:
                    125: @example
                    126: [0] load("gr")$
                    127: [1] load("sp")$
                    128: @end example
                    129:
                    130: @noindent
                    131: $B$"$k$$$O(B, $B>o$KMQ$$$k$J$i$P(B, @samp{$HOME/.asirrc} $B$K=q$$$F$*$/$N$b$h$$(B.
                    132:
                    133: @noindent
                    134: @code{root} $B$O(B $B$=$NB>$N?t$HF1MM(B, $B;MB'1i;;$,2DG=$H$J$k(B. $B$7$+$7(B, $BDj5AB?(B
                    135: $B9`<0$K$h$k4JC12=$O<+F0E*$K$O9T$o$l$J$$$N$G(B, $B%f!<%6$NH=CG$GE,599T$o(B
                    136: $B$J$1$l$P$J$i$J$$(B. $BFC$K(B, $BJ,Jl$,(B 0 $B$K$J$k>l9g$KCWL?E*$J%(%i!<$H$J$k$?$a(B,
                    137: $B<B:]$KJ,Jl$r;}$DBe?tE*?t$r@8@.$9$k>l9g$K$O:Y?4$NCm0U$,I,MW$H$J$k(B.
                    138:
                    139: @noindent
                    140: $BBe?tE*?t$N(B, $BDj5AB?9`<0$K$h$k4JC12=$O(B, @code{simpalg()} $B$G9T$&(B.
                    141:
                    142: @example
                    143: [49] T=A0^2+1;
                    144: (#0^2+1)
                    145: [50] simpalg(T);
                    146: 0
                    147: @end example
                    148:
                    149: @noindent
                    150: @code{simpalg()} $B$OM-M}<0$N7A$r$7$?Be?tE*?t$r(B, $BB?9`<0$N7A$K4JC12=$9$k(B.
                    151:
                    152: @example
                    153: [39] A0=newalg(x^2+1);
                    154: (#0)
                    155: [40] T=(A0^2+A0+1)/(A0+3);
                    156: ((#0^2+#0+1)/(#0+3))
                    157: [41] simpalg(T);
                    158: (3/10*#0+1/10)
                    159: [42] T=1/(A0^2+1);
                    160: ((1)/(#0^2+1))
                    161: [43] simpalg(T);
                    162: div : division by 0
                    163: stopped in invalgp at line 258 in file "/usr/local/lib/asir/sp"
                    164: 258                     return 1/A;
                    165: (debug)
                    166: @end example
                    167:
                    168: @noindent
                    169: $B$3$NNc$G$O(B, $BJ,Jl$,(B 0 $B$NBe?tE*?t$r4JC12=$7$h$&$H$7$F(B 0 $B$K$h$k=|;;$,@8$8(B
                    170: $B$?$?$a(B, $B%f!<%6Dj5AH!?t$G$"$k(B @code{simpalg()} $B$NCf$G%G%P%C%,$,8F$P$l$?(B
                    171: $B$3$H$r<($9(B. @code{simpalg()} $B$O(B, $BBe?tE*?t$r78?t$H$9$kB?9`<0$N(B
                    172: $B3F78?t$r4JC12=$G$-$k(B.
                    173:
                    174: @example
                    175: [43] simpalg(1/A0*x+1/(A0+1));
                    176: (-#0)*x+(-1/2*#0+1/2)
                    177: @end example
                    178:
                    179: @noindent
                    180: $BBe?tE*?t$r78?t$H$9$kB?9`<0$N4pK\1i;;$O(B, $BE,59(B @code{simpalg()} $B$r8F$V$3$H$r(B
                    181: $B=|$1$PDL>o$N>l9g$HF1MM$G$"$k$,(B, $B0x?tJ,2r$J$I$GIQHK$KMQ$$$i$l$k%N%k%`$N(B
                    182: $B7W;;$J$I$K$*$$$F$O(B, @code{root} $B$rITDj85$KCV$-49$($kI,MW$,=P$F$/$k(B.
                    183: $B$3$N>l9g(B, @code{algptorat()} $B$rMQ$$$k(B.
                    184:
                    185: @example
                    186: [83] A0=newalg(x^2+1);
                    187: (#0)
                    188: [84] A1=newalg(x^3+A0*x+A0);
                    189: (#1)
                    190: [85] T=(2*A0+A1*A0+A1^2)*x+(1+A1)/(2+A0);
                    191: (#1^2+#0*#1+2*#0)*x+((#1+1)/(#0+2))
                    192: [86] S=algptorat(T);
                    193: (((t#0+2)*t#1^2+(t#0^2+2*t#0)*t#1+2*t#0^2+4*t#0)*x+t#1+1)/(t#0+2)
                    194: [87] algptorat(coef(T,1));
                    195: t#1^2+t#0*t#1+2*t#0
                    196: @end example
                    197:
                    198: @noindent
                    199: $B$3$N$h$&$K(B, @code{algptorat()} $B$O(B, $BB?9`<0(B, $B?t$K4^$^$l$k(B @code{root}
                    200: $B$r(B, $BBP1~$9$kITDj85(B, $B$9$J$o$A(B @code{#@var{n}} $B$KBP$9$k(B @code{t#@var{n}}
                    201: $B$KCV$-49$($k(B. $B4{$K=R$Y$?$h$&$K(B, $B$3$NITDj85$O%f!<%6$,F~NO$9$k$3$H$O$G$-$J$$(B.
                    202: $B$3$l$O(B, $B%f!<%6$NF~NO$7$?ITDj85$H(B, @code{root} $B$KBP1~$9$kITDj85$,0lCW(B
                    203: $B$7$J$$$h$&$K$9$k$?$a$G$"$k(B.
                    204:
                    205: @noindent
                    206: $B5U$K(B, @code{root} $B$KBP1~$9$kITDj85$r(B, $BBP1~$9$k(B @code{root} $B$KCV$-49$($k(B
                    207: $B$?$a$K$O(B @code{rattoalgp()} $B$rMQ$$$k(B.
                    208:
                    209: @example
                    210: [88] rattoalgp(S,[alg(0)]);
                    211: (((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1+((2*#0^2+4*#0)/(#0+2)))*x
                    212: +((1)/(#0+2))*t#1+((1)/(#0+2))
                    213: [89] rattoalgp(S,[alg(0),alg(1)]);
                    214: (((#0^3+6*#0^2+12*#0+8)*#1^2+(#0^4+6*#0^3+12*#0^2+8*#0)*#1+2*#0^4+12*#0^3
                    215: +24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x+(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))
                    216: [90] rattoalgp(S,[alg(1),alg(0)]);
                    217: (((#0+2)*#1^2+(#0^2+2*#0)*#1+2*#0^2+4*#0)/(#0+2))*x+((#1+1)/(#0+2))
                    218: [91] simpalg(@@89);
                    219: (#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)
                    220: [92] simpalg(@@90);
                    221: (#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)
                    222: @end example
                    223:
                    224: @noindent
                    225: @code{rattoalgp()} $B$O(B, $BCV49$NBP>]$H$J$k(B @code{root} $B$N%j%9%H$rBh(B 2 $B0z?t(B
                    226: $B$K$H$j(B, $B:8$+$i=g$K(B, $BBP1~$9$kITDj85$rCV$-49$($F9T$/(B. $B$3$NNc$O(B,
                    227: $BCV49$9$k=g=x$r49$($k$H4JC12=$r9T$o$J$$$3$H$K$h$j7k2L$,0l8+0[$J$k$,(B,
                    228: $B4JC12=$K$h$j<B$O0lCW$9$k$3$H$r<($7$F$$$k(B. @code{algptorat()},
                    229: @code{rattoalgp()} $B$O(B, $B%f!<%6$,FH<+$N4JC12=$r9T$$$?$$>l9g$J$I$K$b(B
                    230: $BMQ$$$k$3$H$,$G$-$k(B.
                    231:
                    232: @node $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
                    233: @section $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B
                    234:
                    235: @noindent
                    236: @samp{sp} $B$G$O(B, 1 $BJQ?tB?9`<0$K8B$j(B, GCD, $B0x?tJ,2r$*$h$S$=$l$i$N1~MQ$H$7$F(B
                    237: $B:G>.J,2rBN$r5a$a$kH!?t$rDs6!$7$F$$$k(B.
                    238:
                    239: @menu
                    240: * GCD::
                    241: * $BL5J?J}J,2r(B $B0x?tJ,2r(B::
                    242: * $B:G>.J,2rBN(B::
                    243: @end menu
                    244:
                    245: @node GCD,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B
                    246: @subsection GCD
                    247:
                    248: @noindent
                    249: $BBe?tBN>e$G$N(B GCD $B$O(B @code{gcda()} $B$K$h$j7W;;$5$l$k(B.
                    250: $B$3$NH!?t$O%b%8%e%i1i;;$*$h$SCf9q>jM>DjM}$K$h$jBe?tBN>e$N(B GCD $B$r(B
                    251: $B7W;;$9$k$b$N$G(B, $BC`<!3HBg$KBP$7$F$bM-8z$G$"$k(B.
                    252:
                    253: @example
                    254: [63] A=newalg(t^9-15*t^6-87*t^3-125);
                    255: (#0)
                    256: [64] B=newalg(75*s^2+(10*A^7-175*A^4-470*A)*s+3*A^8-45*A^5-261*A^2);
                    257: (#1)
                    258: [65] P1=75*x^2+(150*B+10*A^7-175*A^4-395*A)*x+(75*B^2+(10*A^7-175*A^4-395*A)*B
                    259: +13*A^8-220*A^5-581*A^2)$
                    260: [66] P2=x^2+A*x+A^2$
                    261: [67] gcda(P1,P2,[B,A]);
                    262: 27*x+((#0^6-19*#0^3-65)*#1-#0^7+19*#0^4+38*#0)
                    263: @end example
                    264:
                    265: @node $BL5J?J}J,2r(B $B0x?tJ,2r(B,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B
                    266: @subsection $BL5J?J}J,2r(B, $B0x?tJ,2r(B
                    267:
                    268: @noindent
                    269: $BL5J?J}J,2r$O(B, $BB?9`<0$H$=$NHyJ,$H$N(B GCD $B$N7W;;$+$i;O$^$k$b$C$H$b0lHLE*$J(B
                    270: $B%"%k%4%j%:%`$r:NMQ$7$F$$$k(B. $BH!?t$O(B @code{asq()} $B$G$"$k(B.
                    271:
                    272: @example
                    273: [116] A=newalg(x^2+x+1);
                    274: (#4)
                    275: [117] T=simpalg((x+A+1)*(x^2-2*A-3)^2*(x^3-x-A)^2);
                    276: x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+20)*x^7+(24*#4-6)*x^6
                    277: +(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3+(#4-23)*x^2+(-21*#4-7)*x
                    278: +(3*#4+8)
                    279: [118] asq(T);
                    280: [[x^5+(-2*#4-4)*x^3+(-#4)*x^2+(2*#4+3)*x+(#4-2),2],[x+(#4+1),1]]
                    281: @end example
                    282:
                    283: @noindent
                    284: $B7k2L$ODL>o$HF1MM$K(B, [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$H$J$k$,(B, $BA4$F$N0x;R(B
                    285: $B$N@Q$O(B, $B$b$H$NB?9`<0$HDj?tG\$N:9$O$"$jF@$k(B. $B$3$l$O(B, $B0x;R$r@0?t78?t$K$7(B
                    286: $B$F8+$d$9$/$9$k$?$a$G(B, $B0x?tJ,2r$G$bF1MM$G$"$k(B.
                    287:
                    288: @noindent
                    289: $BBe?tBN>e$G$N0x?tJ,2r$O(B, Trager $B$K$h$k%N%k%`K!$r2~NI$7$?$b$N$G(B, $BFC$K(B
                    290: $B$"$kB?9`<0$KBP$7(B, $B$=$N:,$rE:2C$7$?BN>e$G$=$NB?9`<0<+?H$r0x?tJ,2r$9$k(B
                    291: $B>l9g$KFC$KM-8z$G$"$k(B.
                    292:
                    293: @example
                    294: [119] af(T,[A]);
                    295: [[x^3-x+(-#4),2],[x^2+(-2*#4-3),2],[x+(#4+1),1]]
                    296: @end example
                    297:
                    298: @noindent
                    299: $B0z?t$O(B 2 $B$D$G(B, $BBh(B 2 $B0z?t$O(B, @code{root} $B$N%j%9%H$G$"$k(B. $B0x?tJ,2r$O(B
                    300: $BM-M}?tBN$K(B, $B$=$l$i$N(B @code{root} $B$rE:2C$7$?BN>e$G9T$o$l$k(B.
                    301: @code{root} $B$N=g=x$K$O@)8B$,$"$k(B. $B$9$J$o$A(B, $B8e$GDj5A$5$l$?$b$N$[$I(B
                    302: $BA0$NJ}$K$3$J$1$l$P(B
                    303: $B$J$i$J$$(B. $BJB$Y49$($O(B, $B<+F0E*$K$O9T$o$l$J$$(B. $B%f!<%6$N@UG$$H$J$k(B.
                    304:
                    305: @noindent
                    306: $B%N%k%`$rMQ$$$?0x?tJ,2r$K$*$$$F$O(B, $B%N%k%`$N7W;;$H@0?t78?t(B 1 $BJQ?tB?9`<0$N(B
                    307: $B0x?tJ,2r$N8zN($,(B, $BA4BN$N8zN($r:81&$9$k(B. $B$3$N$&$A(B, $BFC$K9b<!$NB?9`<0(B
                    308: $B$N>l9g$K8e<T$K$*$$$FAH9g$;GzH/$K$h$j7W;;ITG=$K$J$k>l9g$,$7$P$7$P@8$:$k(B.
                    309:
                    310: @example
                    311: [120] B=newalg(x^2-2*A-3);
                    312: (#5)
                    313: [121] af(T,[B,A]);
                    314: [[x+(#5),2],[x^3-x+(-#4),2],[x+(-#5),2],[x+(#4+1),1]]
                    315: @end example
                    316:
                    317: @node $B:G>.J,2rBN(B,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B
                    318: @subsection $B:G>.J,2rBN(B
                    319:
                    320: @noindent
                    321: $B$d$dFC<l$J1i;;$G$O$"$k$,(B, $BA0@a$N0x?tJ,2r$rH?I|E,MQ$9$k$3$H$K$h$j(B,
                    322: $BB?9`<0$N:G>.J,2rBN$r5a$a$k$3$H$,$G$-$k(B. $BH!?t$O(B @code{sp()} $B$G$"$k(B.
                    323:
                    324: @example
                    325: [103] sp(x^5-2);
                    326: [[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),2*x
                    327: +(-#0^3*#1^3),x+(-#0)],[[(#1),t#1^4+t#0*t#1^3+t#0^2*t#1^2+t#0^3*t#1+t#0^4],
                    328: [(#0),t#0^5-2]]]
                    329: @end example
                    330:
                    331: @noindent
                    332: @code{sp()} $B$O(B 1 $B0z?t$G(B, $B7k2L$O(B @code{[1 $B<!0x;R$N%j%9%H(B, [[root,
                    333: algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%j%9%H$G$"$k(B.
                    334: $BBh(B 2 $BMWAG$N(B @code{[root,algptorat($BDj5AB?9`<0(B)]} $B$N%j%9%H$O(B,
                    335: $B1&$+$i=g$K(B, $B:G>.J,2rBN$,F@$i$l$k$^$GE:2C$7$F$$$C$?(B @code{root} $B$r<($9(B.
                    336: $B$=$NDj5AB?9`<0$O(B, $B$=$ND>A0$^$G$N(B @code{root} $B$rE:2C$7$?BN>e$G4{Ls$G$"$k$3$H(B
                    337: $B$,J]>Z$5$l$F$$$k(B.
                    338:
                    339: @noindent
                    340: $B7k2L$NBh(B 1 $BMWAG$G$"$k(B 1 $B<!0x;R$N%j%9%H$O(B, $BBh(B 2 $BMWAG$N(B @code{root} $B$rA4$F(B
                    341: $BE:2C$7$?BN>e$G$N(B, @code{sp()} $B$N0z?t$NB?9`<0$NA4$F$N0x;R$rI=$9(B. $B$=$NBN$O(B
                    342: $B:G>.J,2rBN$H$J$C$F$$$k$N$G(B, $B0x;R$OA4$F(B 1 $B<!$H$J$k$o$1$G$"$k(B. @code{af()}
                    343: $B$HF1MM(B, $BA4$F$N0x;R$N@Q$O(B, $B$b$H$NB?9`<0$HDj?tG\$N:9$O$"$jF@$k(B.
                    344:
                    345:
                    346: @node $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
                    347: @section $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    348:
                    349: @menu
                    350: * newalg::
                    351: * defpoly::
                    352: * alg::
                    353: * algv::
                    354: * simpalg::
                    355: * algptorat::
                    356: * rattoalgp::
                    357: * gcda::
                    358: * sp_norm::
                    359: * asq af::
                    360: * sp::
                    361: @end menu
                    362:
                    363: @node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    364: @subsection @code{newalg}
                    365: @findex newalg
                    366:
                    367: @table @t
                    368: @item newalg(@var{defpoly})
                    369: :: @code{root} $B$r@8@.$9$k(B.
                    370: @end table
                    371:
                    372: @table @var
                    373: @item return
                    374: $BBe?tE*?t(B (@code{root})
                    375: @item defpoly
                    376: $BB?9`<0(B
                    377: @end table
                    378:
                    379: @itemize @bullet
                    380: @item
                    381: @var{defpoly} $B$rDj5AB?9`<0$H$9$kBe?tE*?t(B (@code{root}) $B$r@8@.$9$k(B.
                    382: @item
                    383: @var{defpoly} $B$KBP$9$k@)8B$K4X$7$F$O(B, @xref{$BBe?tE*?t$NI=8=(B}.
                    384: @end itemize
                    385:
                    386: @example
                    387: [0] A0=newalg(x^2-2);
                    388: (#0)
                    389: @end example
                    390:
                    391: @table @t
                    392: @item $B;2>H(B
                    393: @fref{defpoly}
                    394: @end table
                    395:
                    396: @node defpoly,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    397: @subsection @code{defpoly}
                    398: @findex defpoly
                    399:
                    400: @table @t
                    401: @item defpoly(@var{alg})
                    402: :: @code{root} $B$NDj5AB?9`<0$rJV$9(B.
                    403: @end table
                    404:
                    405: @table @var
                    406: @item return
                    407: $BB?9`<0(B
                    408: @item alg
                    409: $BBe?tE*?t(B (@code{root})
                    410: @end table
                    411:
                    412: @itemize @bullet
                    413: @item
                    414: @code{root} @var{alg} $B$NDj5AB?9`<0$rJV$9(B.
                    415: @item
                    416: @code{root} $B$r(B @code{#@var{n}} $B$H$9$l$P(B, $BDj5AB?9`<0$N<gJQ?t$O(B
                    417: @code{t#@var{n}} $B$H$J$k(B.
                    418: @end itemize
                    419:
                    420: @example
                    421: [1] defpoly(A0);
                    422: t#0^2-2
                    423: @end example
                    424:
                    425: @table @t
                    426: @item $B;2>H(B
                    427: @fref{newalg}, @fref{alg}, @fref{algv}
                    428: @end table
                    429:
                    430: @node alg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    431: @subsection @code{alg}
                    432: @findex alg
                    433:
                    434: @table @t
                    435: @item alg(@var{i})
                    436: :: $B%$%s%G%C%/%9$KBP1~$9$k(B @code{root} $B$rJV$9(B.
                    437: @end table
                    438:
                    439: @table @var
                    440: @item return
                    441: $BBe?tE*?t(B (@code{root})
                    442: @item i
                    443: $B@0?t(B
                    444: @end table
                    445:
                    446: @itemize @bullet
                    447: @item
                    448: @code{root} @code{#@var{i}} $B$rJV$9(B.
                    449: @item
                    450: @code{#@var{i}} $B$O%f!<%6$,D>@\F~NO$G$-$J$$$?$a(B, @code{alg(@var{i})} $B$H(B
                    451: $B$$$&7A$GF~NO$9$k(B.
                    452: @end itemize
                    453:
                    454: @example
                    455: [2] x+#0;
                    456: syntax error
                    457: 0
                    458: [3] alg(0);
                    459: (#0)
                    460: @end example
                    461:
                    462: @table @t
                    463: @item $B;2>H(B
                    464: @fref{newalg}, @fref{algv}
                    465: @end table
                    466:
                    467: @node algv,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    468: @subsection @code{algv}
                    469: @findex algv
                    470:
                    471: @table @t
                    472: @item algv(@var{i})
                    473: :: @code{alg(@var{i})} $B$KBP1~$9$kITDj85$rJV$9(B.
                    474: @end table
                    475:
                    476: @table @var
                    477: @item return
                    478: $BB?9`<0(B
                    479: @item i
                    480: $B@0?t(B
                    481: @end table
                    482:
                    483: @itemize @bullet
                    484: @item
                    485: $BB?9`<0(B @code{t#@var{i}} $B$rJV$9(B.
                    486: @item
                    487: @code{t#@var{i}} $B$O%f!<%6$,D>@\F~NO$G$-$J$$$?$a(B, @code{algv(@var{i})} $B$H(B
                    488: $B$$$&7A$GF~NO$9$k(B.
                    489: @end itemize
                    490:
                    491: @example
                    492: [4] var(defpoly(A0));
                    493: t#0
                    494: [5] t#0;
                    495: syntax error
                    496: 0
                    497: [6] algv(0);
                    498: t#0
                    499: @end example
                    500:
                    501: @table @t
                    502: @item $B;2>H(B
                    503: @fref{newalg}, @fref{defpoly}, @fref{alg}
                    504: @end table
                    505:
                    506: @node simpalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    507: @subsection @code{simpalg}
                    508: @findex simpalg
                    509:
                    510: @table @t
                    511: @item simpalg(@var{rat})
                    512: :: $BM-M}<0$K4^$^$l$kBe?tE*?t$r4JC12=$9$k(B.
                    513: @end table
                    514:
                    515: @table @var
                    516: @item return
                    517: $BM-M}<0(B
                    518: @item rat
                    519: $BM-M}<0(B
                    520: @end table
                    521:
                    522: @itemize @bullet
                    523: @item
                    524: @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    525: @item
                    526: $B?t(B, $BB?9`<0(B, $BM-M}<0$K4^$^$l$kBe?tE*?t$r(B, $B4^$^$l$k(B @code{root} $B$NDj5A(B
                    527: $BB?9`<0$K$h$j4JC12=$9$k(B.
                    528: @item
                    529: $B?t$N>l9g(B, $BJ,Jl$,$"$l$PM-M}2=$5$l(B, $B7k2L$O(B @code{root} $B$NB?9`<0$H$J$k(B.
                    530: @item
                    531: $BB?9`<0$N>l9g(B, $B3F78?t$,4JC12=$5$l$k(B.
                    532: @item
                    533: $BM-M}<0$N>l9g(B, $BJ,JlJ,;R$,B?9`<0$H$7$F4JC12=$5$l$k(B.
                    534: @end itemize
                    535:
                    536: @example
                    537: [7] simpalg((1+A0)/(1-A0));
                    538: simpalg undefined
                    539: return to toplevel
                    540: [7] load("sp")$
                    541: [46] simpalg((1+A0)/(1-A0));
                    542: (-2*#0-3)
                    543: [47] simpalg((2-A0)/(2+A0)*x^2-1/(3+A0));
                    544: (-2*#0+3)*x^2+(1/7*#0-3/7)
                    545: [48] simpalg((x+1/(A0-1))/(x-1/(A0+1)));
                    546: (x+(#0+1))/(x+(-#0+1))
                    547: @end example
                    548:
                    549: @node algptorat,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    550: @subsection @code{algptorat}
                    551: @findex algptorat
                    552:
                    553: @table @t
                    554: @item algptorat(@var{poly})
                    555: :: $BB?9`<0$K4^$^$l$k(B @code{root} $B$r(B, $BBP1~$9$kITDj85$KCV$-49$($k(B.
                    556: @end table
                    557:
                    558: @table @var
                    559: @item return
                    560: $BB?9`<0(B
                    561: @item poly
                    562: $BB?9`<0(B
                    563: @end table
                    564:
                    565: @itemize @bullet
                    566: @item
                    567: @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    568: @item
                    569: $BB?9`<0$K4^$^$l$k(B @code{root} @code{#@var{n}} $B$rA4$F(B @code{t#@var{n}} $B$K(B
                    570: $BCV$-49$($k(B.
                    571: @end itemize
                    572:
                    573: @example
                    574: [49] algptorat((-2*alg(0)+3)*x^2+(1/7*alg(0)-3/7));
                    575: (-2*t#0+3)*x^2+1/7*t#0-3/7
                    576: @end example
                    577:
                    578: @table @t
                    579: @item $B;2>H(B
                    580: @fref{defpoly}, @fref{algv}
                    581: @end table
                    582:
                    583: @node rattoalgp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    584: @subsection @code{rattoalgp}
                    585: @findex rattoalgp
                    586:
                    587: @table @t
                    588: @item rattoalgp(@var{poly},@var{alglist})
                    589: :: $BB?9`<0$K4^$^$l$k(B @code{root} $B$KBP1~$9$kITDj85$r(B @code{root} $B$K(B
                    590: $BCV$-49$($k(B.
                    591: @end table
                    592:
                    593: @table @var
                    594: @item return
                    595: $BB?9`<0(B
                    596: @item poly
                    597: $BB?9`<0(B
                    598: @item alglist
                    599: $B%j%9%H(B
                    600: @end table
                    601:
                    602: @itemize @bullet
                    603: @item
                    604: @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    605: @item
                    606: $BBh(B 2 $B0z?t$O(B @code{root} $B$N%j%9%H$G$"$k(B. @code{rattoalgp()} $B$O(B, $B$3$N(B @code{root}
                    607: $B$KBP1~$9$kITDj85$r(B, $B$=$l$>$l(B @code{root} $B$KCV$-49$($k(B.
                    608: @end itemize
                    609:
                    610: @example
                    611: [51] rattoalgp((-2*algv(0)+3)*x^2+(1/7*algv(0)-3/7),[alg(0)]);
                    612: (-2*#0+3)*x^2+(1/7*#0-3/7)
                    613: @end example
                    614:
                    615: @table @t
                    616: @item $B;2>H(B
                    617: @fref{alg}, @fref{algv}
                    618: @end table
                    619:
                    620: @node gcda,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    621: @subsection @code{gcda}
                    622: @findex gcda
                    623:
                    624: @table @t
                    625: @item gcda(@var{poly1},@var{poly2},@var{alist})
                    626: :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N(B GCD
                    627: @end table
                    628:
                    629: @table @var
                    630: @item return
                    631: $BB?9`<0(B
                    632: @item poly1, poly2
                    633: $BB?9`<0(B
                    634: @item alist
                    635: $B%j%9%H(B
                    636: @end table
                    637:
                    638: @itemize @bullet
                    639: @item
                    640: @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    641: @item
                    642: 2 $B$D$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$r5a$a$k(B.
                    643: @item
                    644: @var{alist} $B$OF~NO$K8=$l$k(B @code{root} $B$*$h$S(B, $B$=$l$i$NDj5A$K4^$^$l$k(B
                    645: @code{root} $B$r:F5"E*$K<h$j=P$7$FJB$Y$?%j%9%H(B. @var{a} $B$,(B @var{b} $B$N(B
                    646: $BDj5A$K4^$^$l$F$$$k>l9g(B, @var{a} $B$O(B @var{b} $B$h$j8e(B ($B1&(B) $B$KJB$P$J$1$l$P(B
                    647: $B$J$i$J$$(B.
                    648: @end itemize
                    649:
                    650: @example
                    651: [76] X=x^6+3*x^5+6*x^4+x^3-3*x^2+12*x+16$
                    652: [77] Y=x^6+6*x^5+24*x^4+8*x^3-48*x^2+384*x+1024$
                    653: [78] A=newalg(X);
                    654: (#0)
                    655: [79] gcda(X,subst(Y,x,x+A),[A]);
                    656: x+(-#0)
                    657: @end example
                    658:
                    659: @table @t
                    660: @item $B;2>H(B
                    661: @fref{gr hgr gr_mod}, @fref{asq af}
                    662: @end table
                    663:
                    664: @node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    665: @subsection @code{sp_norm}
                    666: @findex sp_norm
                    667:
                    668: @table @t
                    669: @item sp_norm(@var{alg},@var{var},@var{poly},@var{alglist})
                    670: :: $BBe?tBN>e$G$N%N%k%`$N7W;;(B
                    671: @end table
                    672:
                    673: @table @var
                    674: @item return
                    675: $BB?9`<0(B
                    676: @item var
                    677: @var{poly} $B$N<gJQ?t(B
                    678: @item poly
                    679: 1 $BJQ?tB?9`<0(B
                    680: @item alg
                    681: @code{root}
                    682: @item alglist
                    683: @code{root} $B$N%j%9%H(B
                    684: @end table
                    685:
                    686: @itemize @bullet
                    687: @item
                    688: @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    689: @item
                    690: @var{poly} $B$N(B, @var{alg} $B$K4X$9$k%N%k%`$r$H$k(B. $B$9$J$o$A(B,
                    691: @b{K} = @b{Q}(@var{alglist} \ @{@var{alg}@}) $B$H$9$k$H$-(B,
                    692: @var{poly} $B$K8=$l$k(B @var{alg} $B$r(B, @var{alg} $B$N(B @b{K} $B>e$N6&Lr$KCV$-49$($?$b$N(B
                    693: $BA4$F$N@Q$rJV$9(B.
                    694: @item
                    695: $B7k2L$O(B @b{K} $B>e$NB?9`<0$H$J$k(B.
                    696: @item
                    697: $B<B:]$K$OF~NO$K$h$j>l9g$o$1$,9T$o$l(B, $B=*7k<0$ND>@\7W;;$dCf9q>jM>DjM}$K(B
                    698: $B$h$j7W;;$5$l$k$,(B, $B:GE,$JA*Br$,9T$o$l$F$$$k$H$O8B$i$J$$(B.
                    699: $BBg0hJQ?t(B @code{USE_RES} $B$r(B 1 $B$K@_Dj$9$k$3$H$K$h$j(B, $B>o$K=*7k<0$K$h$j7W;;(B
                    700: $B$5$;$k$3$H$,$G$-$k(B.
                    701: @end itemize
                    702:
                    703: @example
                    704: [0] load("sp")$
                    705: [39] A0=newalg(x^2+1)$
                    706: [40] A1=newalg(x^2+A0)$
                    707: [41] sp_norm(A1,x,x^3+A0*x+A1,[A1,A0]);
                    708: x^6+(2*#0)*x^4+(#0^2)*x^2+(#0)
                    709: [42] sp_norm(A0,x,@@@@,[A0]);
                    710: x^12+2*x^8+5*x^4+1
                    711: @end example
                    712:
                    713: @table @t
                    714: @item $B;2>H(B
                    715: @fref{res}, @fref{asq af}
                    716: @end table
                    717:
                    718: @node asq af,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    719: @subsection @code{asq}, @code{af}
                    720: @findex asq
                    721: @findex af
                    722:
                    723: @table @t
                    724: @item asq(@var{poly})
                    725: :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$NL5J?J}J,2r(B
                    726: @item af(@var{poly},@var{alglist})
                    727: :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
                    728: @end table
                    729:
                    730: @table @var
                    731: @item return
                    732: $B%j%9%H(B
                    733: @item poly
                    734: $BB?9`<0(B
                    735: @item alglist
                    736: @code{root} $B$N%j%9%H(B
                    737: @end table
                    738:
                    739: @itemize @bullet
                    740: @item
                    741: $B$$$:$l$b(B @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    742: @item
                    743: @code{root} $B$r4^$^$J$$>l9g$O@0?t>e$NH!?t$,8F$S=P$5$l9bB.$G$"$k$,(B,
                    744: @code{root} $B$r4^$`>l9g$K$O(B, @code{gcda()} $B$,5/F0$5$l$k$?$a$7$P$7$P(B
                    745: $B;~4V$,$+$+$k(B.
                    746: @item
                    747: @code{af()} $B$O(B, $B4pACBN$N;XDj(B, $B$9$J$o$ABh(B 2 $B0z?t$N(B, @code{root} $B$N%j%9%H(B
                    748: $B$N;XDj$,I,MW$G$"$k(B.
                    749: @item
                    750: @code{alglist} $B$G;XDj$5$l$k(B @code{root} $B$O(B, $B8e$GDj5A$5$l$?$b$N$[$IA0$N(B
                    751: $BJ}$KMh$J$1$l$P$J$i$J$$(B.
                    752: @item
                    753: $B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$G$"$k(B.
                    754: @item
                    755: $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B.
                    756: @end itemize
                    757:
                    758: @example
                    759: [99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2);
                    760: [[-x^2+3*x+(#0),2]]
                    761: [100] af(-x^2+3*x+alg(0),[alg(0)]);
                    762: [[x+(#0-1),1],[-x+(#0+2),1]]
                    763: @end example
                    764:
                    765: @table @t
                    766: @item $B;2>H(B
                    767: @fref{gcda}, @fref{fctr sqfr}
                    768: @end table
                    769:
                    770: @node sp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
                    771: @subsection @code{sp}
                    772: @findex sp
                    773:
                    774: @table @t
                    775: @item sp(@var{poly})
                    776: :: $B:G>.J,2rBN$r5a$a$k(B.
                    777: @end table
                    778:
                    779: @table @var
                    780: @item return
                    781: $B%j%9%H(B
                    782: @item poly
                    783: $BB?9`<0(B
                    784: @end table
                    785:
                    786: @itemize @bullet
                    787: @item
                    788: @samp{sp} $B$GDj5A$5$l$F$$$k(B.
                    789: @item
                    790: $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B @var{poly} $B$N:G>.J,2rBN(B, $B$*$h$S$=$NBN>e$G$N(B
                    791: @var{poly} $B$N(B 1 $B<!0x;R$X$NJ,2r$r5a$a$k(B.
                    792: @item
                    793: $B7k2L$O(B, @var{poly} $B$N0x;R$N%j%9%H$H(B, $B:G>.J,2rBN$N(B, $BC`<!3HBg$K$h$kI=8=(B
                    794: $B$+$i$J$k%j%9%H$G$"$k(B.
                    795: @item
                    796: $B:G>.J,2rBN$O(B, @code{[root,algptorat(defpoly(root))]} $B$N%j%9%H$H$7$F(B
                    797: $BI=8=$5$l$F$$$k(B. $B$9$J$o$A(B, $B5a$a$k:G>.J,2rBN$O(B, $BM-M}?tBN$K(B, $B$3$N(B @code{root}
                    798: $B$rA4$FE:2C$7$?BN$H$7$FF@$i$l$k(B. $BE:2C$O(B, $B1&$NJ}$N(B @code{root} $B$+$i=g$K(B
                    799: $B9T$o$l$k(B.
                    800: @item
                    801: @code{sp()} $B$O(B, $BFbIt$G%N%k%`$N7W;;$N$?$a$K(B @code{sp_norm()} $B$r$7$P$7$P(B
                    802: $B5/F0$9$k(B. $B%N%k%`$N7W;;$O(B, $B>u67$K1~$8$F$5$^$6$^$JJ}K!$G9T$o$l$k$,(B,
                    803: $B$=$3$GMQ$$$i$l$kJ}K!$,:GA1$H$O8B$i$:(B, $BC1=c$J=*7k<0$N7W;;$NJ}$,9bB.(B
                    804: $B$G$"$k>l9g$b$"$k(B.
                    805: $BBg0hJQ?t(B @code{USE_RES} $B$r(B 1 $B$K@_Dj$9$k$3$H$K$h$j(B, $B>o$K=*7k<0$K$h$j7W;;(B
                    806: $B$5$;$k$3$H$,$G$-$k(B.
                    807: @end itemize
                    808:
                    809: @example
                    810: [101] L=sp(x^9-54);
                    811: [[x+(-#2),-54*x+(#1^6*#2^4),54*x+(#1^6*#2^4+54*#2),54*x+(-#1^8*#2^2),
                    812: -54*x+(#1^5*#2^5),54*x+(#1^5*#2^5+#1^8*#2^2),-54*x+(-#1^7*#2^3-54*#1),
                    813: 54*x+(-#1^7*#2^3),x+(-#1)],[[(#2),t#2^6+t#1^3*t#2^3+t#1^6],[(#1),t#1^9-54]]]
                    814: [102] for(I=0,M=1;I<9;I++)M*=L[0][I];
                    815: [111] M=simpalg(M);
                    816: -1338925209984*x^9+72301961339136
                    817: [112] ptozp(M);
                    818: -x^9+54
                    819: @end example
                    820:
                    821: @table @t
                    822: @item $B;2>H(B
                    823: @fref{asq af}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}.
                    824: @end table
                    825: