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