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Diff for /OpenXM/src/asir-doc/parts/algnum.texi between version 1.4 and 1.7

version 1.4, 2000/03/17 02:17:03

version 1.7, 2003/04/20 08:01:24

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/algnum.texi,v 1.3 2000/03/10 07:18:40 noro Exp $

@comment $OpenXM: OpenXM/src/asir-doc/parts/algnum.texi,v 1.6 2003/04/19 15:44:55 noro Exp $

\BJP

@node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top

@chapter $BBe?tE*?t$K4X$9$k1i;;(B

Line 410 into the @b{root} by @code{rattoalgp()} function.

@example

[88] rattoalgp(S,[alg(0)]);

(((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1+((2*#0^2+4*#0)/(#0+2)))*x

(((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1

+((1)/(#0+2))*t#1+((1)/(#0+2))

+((2*#0^2+4*#0)/(#0+2)))*x+((1)/(#0+2))*t#1+((1)/(#0+2))

[89] rattoalgp(S,[alg(0),alg(1)]);

(((#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

(((#0^3+6*#0^2+12*#0+8)*#1^2+(#0^4+6*#0^3+12*#0^2+8*#0)*#1

+24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x+(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))

+2*#0^4+12*#0^3+24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x

+(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))

[90] rattoalgp(S,[alg(1),alg(0)]);

(((#0+2)*#1^2+(#0^2+2*#0)*#1+2*#0^2+4*#0)/(#0+2))*x+((#1+1)/(#0+2))

(((#0+2)*#1^2+(#0^2+2*#0)*#1+2*#0^2+4*#0)/(#0+2))*x

+((#1+1)/(#0+2))

[91] simpalg(@@89);

(#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)

[92] simpalg(@@90);

Line 499 where the ground field is a multiple extension.

Line 501 where the ground field is a multiple extension.

(#0)

[64] B=newalg(75*s^2+(10*A^7-175*A^4-470*A)*s+3*A^8-45*A^5-261*A^2);

(#1)

[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

[65] P1=75*x^2+(150*B+10*A^7-175*A^4-395*A)*x

+13*A^8-220*A^5-581*A^2)$

+(75*B^2+(10*A^7-175*A^4-395*A)*B+13*A^8-220*A^5-581*A^2)$

[66] P2=x^2+A*x+A^2$

[67] cr_gcda(P1,P2);

27*x+((#0^6-19*#0^3-65)*#1-#0^7+19*#0^4+38*#0)

Line 531 The function to do this factorization is @code{asq()}.

Line 533 The function to do this factorization is @code{asq()}.

[116] A=newalg(x^2+x+1);

(#4)

[117] T=simpalg((x+A+1)*(x^2-2*A-3)^2*(x^3-x-A)^2);

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

x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+20)*x^7

+(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3+(#4-23)*x^2+(-21*#4-7)*x

+(24*#4-6)*x^6+(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3

+(3*#4+8)

+(#4-23)*x^2+(-21*#4-7)*x+(3*#4+8)

[118] asq(T);

[[x^5+(-2*#4-4)*x^3+(-#4)*x^2+(2*#4+3)*x+(#4-2),2],[x+(#4+1),1]]

@end example

Line 641 The function is @code{sp()}.

Line 643 The function is @code{sp()}.

@example

[103] sp(x^5-2);

[[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),2*x

[[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),

+(-#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],

2*x+(-#0^3*#1^3),x+(-#0)],

[(#0),t#0^5-2]]]

[[(#1),t#1^4+t#0*t#1^3+t#0^2*t#1^2+t#0^3*t#1+t#0^4],[(#0),t#0^5-2]]]

@end example

@noindent

Line 1083 substitutes a @b{root} for the associated indeterminat

Line 1085 substitutes a @b{root} for the associated indeterminat

@item return

\JP $BB?9`<0(B

\EG polynomial

@item poly1, poly2

@item poly1 poly2

\JP $BB?9`<0(B

\EG polynomial

@end table

Line 1269 first.

Line 1271 first.

@item

\BJP

@code{sp_noalg} $B$G$O(B, @var{poly} $B$K4^$^$l$kBe?tE*?t(B @var{ai} $B$rITDj85(B @var{vi}

@code{af(F,AL)} $B$K$*$$$F(B, @code{AL} $B$OBe?tE*?t$N%j%9%H$G$"$j(B, $BM-M}?tBN$N(B

$B$GCV$-49$($k(B. @code{defpolylist} $B$O(B, @var{[[vn,dn(vn,...,v1)],...,[v1,d(v1)]]}

$BBe?t3HBg$rI=$9(B. @code{AL=[An,...,A1]} $B$H=q$/$H$-(B, $B3F(B @code{Ak} $B$O(B, $B$=$l$h$j(B

$B$J$k%j%9%H$G$"$k(B. $B$3$3$G(B @var{di(vi,...,v1)} $B$O(B @var{ai} $B$NDj5AB?9`<0$K$*$$$F(B

$B1&$K$"$kBe?tE*?t$r78?t$H$7$?(B, $B%b%K%C%/$JDj5AB?9`<0$GDj5A$5$l$F$$$J$1$l$P(B

$B$J$i$J$$(B.

\BEG

In @code{af(F,AL)}, @code{AL} denotes a list of @code{roots} and it

represents an algebraic number field. In @code{AL=[An,...,A1]} each

@code{Ak} should be defined as a root of a defining polynomial

whose coefficients are in @code{Q(A(k+1),...,An)}.

@example

[1] A1 = newalg(x^2+1);

[2] A2 = newalg(x^2+A1);

[3] A3 = newalg(x^2+A2*x+A1);

[4] af(x^2+A2*x+A1,[A2,A1]);

[[x^2+(#1)*x+(#0),1]]

@end example

\BJP

@code{af_noalg} $B$G$O(B, @var{poly} $B$K4^$^$l$kBe?tE*?t(B @var{ai} $B$rITDj85(B @var{vi}

$B$GCV$-49$($k(B. @code{defpolylist} $B$O(B, [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]

$B$J$k%j%9%H$G$"$k(B. $B$3$3$G(B @var{di}(vi,...,v1) $B$O(B @var{ai} $B$NDj5AB?9`<0$K$*$$$F(B

$BBe?tE*?t$rA4$F(B @var{vj} $B$KCV$-49$($?$b$N$G$"$k(B.

\BEG

To call @code{sp_noalg}, one should replace each algebraic number

@var{ai} in @var{poly} with an indeterminate @var{vi}. @code{defpolylist}

is a list @var{[[vn,dn(vn,...,v1)],...,[v1,d(v1)]]}. In this expression

is a list [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]. In this expression

@var{di(vi,...,v1)} is a defining polynomial of @var{ai} represented

@var{di}(vi,...,v1) is a defining polynomial of @var{ai} represented

as a multivariate polynomial.

@example

[1] af_noalg(x^2+a2*x+a1,[[a2,a2^2+a1],[a1,a1^2+1]]);

[[x^2+a2*x+a1,1]]

@end example

@item

\BJP

$B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}]

Line 1302 the input polynomial by a constant.

Line 1331 the input polynomial by a constant.

@end itemize

@example

[98] A = newalg(t^2-2);

(#0)

[99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2);

[[-x^2+3*x+(#0),2]]

[100] af(-x^2+3*x+alg(0),[alg(0)]);

[[x+(#0-1),1],[-x+(#0+2),1]]

[101] af_noalg(-x^2+3*x+a,[[a,x^2-2]]);

[[x+a-1,1],[-x+a+2,1]]

@end example

@table @t

Line 1374 is a list containing only integral polynomials.

Line 1407 is a list containing only integral polynomials.

\BEG

The splitting field is represented as a list of pairs of form

@code{[root,algptorat(defpoly(root))]}.

@code{[root,} @code{algptorat(defpoly(root))]}.

In more detail, the list is interpreted as a representation

of successive extension obtained by adjoining @b{root}'s

to the rational number field. Adjoining is performed from the right

Line 1401 the builtin function @code{res()} is always used.

Line 1434 the builtin function @code{res()} is always used.

@example

[101] L=sp(x^9-54);

[[x+(-#2),-54*x+(#1^6*#2^4),54*x+(#1^6*#2^4+54*#2),54*x+(-#1^8*#2^2),

[[x+(-#2),-54*x+(#1^6*#2^4),54*x+(#1^6*#2^4+54*#2),

-54*x+(#1^5*#2^5),54*x+(#1^5*#2^5+#1^8*#2^2),-54*x+(-#1^7*#2^3-54*#1),

54*x+(-#1^8*#2^2),-54*x+(#1^5*#2^5),54*x+(#1^5*#2^5+#1^8*#2^2),

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]]]

-54*x+(-#1^7*#2^3-54*#1),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]]]

[102] for(I=0,M=1;I<9;I++)M*=L[0][I];

[111] M=simpalg(M);

-1338925209984*x^9+72301961339136

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