| version 1.3, 2000/03/10 07:18:40 |
version 1.5, 2000/09/23 07:53:24 |
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| @comment $OpenXM: OpenXM/src/asir-doc/parts/algnum.texi,v 1.2 1999/12/21 02:47:30 noro Exp $ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/algnum.texi,v 1.4 2000/03/17 02:17:03 noro Exp $ |
| \BJP |
\BJP |
| @node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top |
@node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top |
| @chapter $BBe?tE*?t$K4X$9$k1i;;(B |
@chapter $BBe?tE*?t$K4X$9$k1i;;(B |
| Line 706 may yield a polynomial which differs by a constant. |
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| Line 706 may yield a polynomial which differs by a constant. |
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| * rattoalgp:: |
* rattoalgp:: |
| * cr_gcda:: |
* cr_gcda:: |
| * sp_norm:: |
* sp_norm:: |
| * asq af:: |
* asq af af_noalg:: |
| * sp:: |
* sp sp_noalg:: |
| @end menu |
@end menu |
| |
|
| \JP @node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
\JP @node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
|
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item Reference |
\EG @item Reference |
| @fref{gr hgr gr_mod}, @fref{asq af} |
@fref{gr hgr gr_mod}, @fref{asq af af_noalg} |
| @end table |
@end table |
| |
|
| \JP @node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
\JP @node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
| Line 1192 x^12+2*x^8+5*x^4+1 |
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| Line 1192 x^12+2*x^8+5*x^4+1 |
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| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item Reference |
\EG @item Reference |
| @fref{res}, @fref{asq af} |
@fref{res}, @fref{asq af af_noalg} |
| @end table |
@end table |
| |
|
| \JP @node asq af,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
\JP @node asq af af_noalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
| \EG @node asq af,,, Summary of functions for algebraic numbers |
\EG @node asq af af_noalg,,, Summary of functions for algebraic numbers |
| @subsection @code{asq}, @code{af} |
@subsection @code{asq}, @code{af}, @code{af_noalg} |
| @findex asq |
@findex asq |
| @findex af |
@findex af |
| |
@findex af_noalg |
| |
|
| @table @t |
@table @t |
| @item asq(@var{poly}) |
@item asq(@var{poly}) |
| Line 1209 x^12+2*x^8+5*x^4+1 |
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| Line 1210 x^12+2*x^8+5*x^4+1 |
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| algebraic number field. |
algebraic number field. |
| \E |
\E |
| @item af(@var{poly},@var{alglist}) |
@item af(@var{poly},@var{alglist}) |
| |
@itemx af_noalg(@var{poly},@var{defpolylist}) |
| \JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B |
\JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B |
| \BEG |
\BEG |
| :: Factorization of polynomial @var{poly} over an |
:: Factorization of polynomial @var{poly} over an |
| Line 1226 algebraic number field. |
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| Line 1228 algebraic number field. |
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| @item alglist |
@item alglist |
| \JP @code{root} $B$N%j%9%H(B |
\JP @code{root} $B$N%j%9%H(B |
| \EG @code{root} list |
\EG @code{root} list |
| |
@item defpolylist |
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\JP @code{root} $B$rI=$9ITDj85$HDj5AB?9`<0$N%Z%"$N%j%9%H(B |
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\EG @code{root} list of pairs of an indeterminate and a polynomial |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| Line 1263 In the second argument @code{alglist}, @b{root} define |
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| Line 1268 In the second argument @code{alglist}, @b{root} define |
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| first. |
first. |
| \E |
\E |
| @item |
@item |
| \JP $B7k2L$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. |
\BJP |
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@code{af(F,AL)} $B$K$*$$$F(B, @code{AL} $B$OBe?tE*?t$N%j%9%H$G$"$j(B, $BM-M}?tBN$N(B |
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$BBe?t3HBg$rI=$9(B. @code{AL=[An,...,A1]} $B$H=q$/$H$-(B, $B3F(B @code{Ak} $B$O(B, $B$=$l$h$j(B |
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$B1&$K$"$kBe?tE*?t$r78?t$H$7$?(B, $B%b%K%C%/$JDj5AB?9`<0$GDj5A$5$l$F$$$J$1$l$P(B |
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$B$J$i$J$$(B. |
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\E |
| \BEG |
\BEG |
| |
In @code{af(F,AL)}, @code{AL} denotes a list of @code{roots} and it |
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represents an algebraic number field. In @code{AL=[An,...,A1]} each |
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@code{Ak} should be defined as a root of a defining polynomial |
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whose coefficients are in @code{Q(A(k+1),...,An)}. |
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\E |
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|
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@example |
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[1] A1 = newalg(x^2+1); |
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[2] A2 = newalg(x^2+A1); |
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[3] A3 = newalg(x^2+A2*x+A1); |
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[4] af(x^2+A2*x+A1,[A2,A1]); |
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[[x^2+(#1)*x+(#0),1]] |
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@end example |
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|
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\BJP |
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@code{af_noalg} $B$G$O(B, @var{poly} $B$K4^$^$l$kBe?tE*?t(B @var{ai} $B$rITDj85(B @var{vi} |
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$B$GCV$-49$($k(B. @code{defpolylist} $B$O(B, @var{[[vn,dn(vn,...,v1)],...,[v1,d(v1)]]} |
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$B$J$k%j%9%H$G$"$k(B. $B$3$3$G(B @var{di(vi,...,v1)} $B$O(B @var{ai} $B$NDj5AB?9`<0$K$*$$$F(B |
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$BBe?tE*?t$rA4$F(B @var{vj} $B$KCV$-49$($?$b$N$G$"$k(B. |
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\E |
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\BEG |
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To call @code{sp_noalg}, one should replace each algebraic number |
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@var{ai} in @var{poly} with an indeterminate @var{vi}. @code{defpolylist} |
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is a list @var{[[vn,dn(vn,...,v1)],...,[v1,d(v1)]]}. In this expression |
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@var{di(vi,...,v1)} is a defining polynomial of @var{ai} represented |
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as a multivariate polynomial. |
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\E |
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|
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@example |
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[1] af_noalg(x^2+a2*x+a1,[[a2,a2^2+a1],[a1,a1^2+1]]); |
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[[x^2+a2*x+a1,1]] |
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@end example |
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|
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@item |
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\BJP |
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$B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] |
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$B$N%j%9%H$G$"$k(B. @code{af_noalg} $B$N>l9g(B, @b{$B0x;R(B} $B$K8=$l$kBe?tE*?t$O(B, |
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@var{defpolylist} $B$K=>$C$FITDj85$KCV$-49$($i$l$k(B. |
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\E |
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\BEG |
| The result is a list, as a result of usual factorization, whose elements |
The result is a list, as a result of usual factorization, whose elements |
| is of the form [@b{factor}, @b{multiplicity}]. |
is of the form [@b{factor}, @b{multiplicity}]. |
| |
In the result of @code{af_noalg}, algebraic numbers in @v{factor} are |
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replaced by the indeterminates according to @var{defpolylist}. |
| \E |
\E |
| @item |
@item |
| \JP $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B. |
\JP $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B. |
| Line 1277 the input polynomial by a constant. |
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| Line 1329 the input polynomial by a constant. |
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| @end itemize |
@end itemize |
| |
|
| @example |
@example |
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[98] A = newalg(t^2-2); |
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(#0) |
| [99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2); |
[99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2); |
| [[-x^2+3*x+(#0),2]] |
[[-x^2+3*x+(#0),2]] |
| [100] af(-x^2+3*x+alg(0),[alg(0)]); |
[100] af(-x^2+3*x+alg(0),[alg(0)]); |
| [[x+(#0-1),1],[-x+(#0+2),1]] |
[[x+(#0-1),1],[-x+(#0+2),1]] |
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[101] af_noalg(-x^2+3*x+a,[[a,x^2-2]]); |
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[[x+a-1,1],[-x+a+2,1]] |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| Line 1289 the input polynomial by a constant. |
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| Line 1345 the input polynomial by a constant. |
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| @fref{cr_gcda}, @fref{fctr sqfr} |
@fref{cr_gcda}, @fref{fctr sqfr} |
| @end table |
@end table |
| |
|
| \JP @node sp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
\JP @node sp sp_noalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B |
| \EG @node sp,,, Summary of functions for algebraic numbers |
\EG @node sp sp_noalg,,, Summary of functions for algebraic numbers |
| @subsection @code{sp} |
@subsection @code{sp}, @code{sp_noalg} |
| @findex sp |
@findex sp |
| |
|
| @table @t |
@table @t |
| @item sp(@var{poly}) |
@item sp(@var{poly}) |
| |
@itemx sp_noalg(@var{poly}) |
| \JP :: $B:G>.J,2rBN$r5a$a$k(B. |
\JP :: $B:G>.J,2rBN$r5a$a$k(B. |
| \EG :: Finds the splitting field of polynomial @var{poly} and splits. |
\EG :: Finds the splitting field of polynomial @var{poly} and splits. |
| @end table |
@end table |
| Line 1326 over the field. |
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| Line 1383 over the field. |
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| @item |
@item |
| \BJP |
\BJP |
| $B7k2L$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 |
$B7k2L$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 |
| $B$+$i$J$k%j%9%H$G$"$k(B. |
$B$+$i$J$k%j%9%H$G$"$k(B. @code{sp_noalg} $B$G$O(B, $BA4$F$NBe?tE*?t$,(B, $BBP1~$9$k(B |
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$BITDj85(B ($BB($A(B @code{#i} $B$KBP$9$k(B @code{t#i}) $B$KCV$-49$($i$l$k(B. $B$3$l$K(B |
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$B$h$j(B, @code{sp_noalg} $B$N=PNO$O(B, $B@0?t78?tB?JQ?tB?9`<0$N%j%9%H$H$J$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| The result consists of a two element list: The first element is |
The result consists of a two element list: The first element is |
| the list of all linear factors of @var{poly}; the second element is |
the list of all linear factors of @var{poly}; the second element is |
| a list which represents the successive extension of the field. |
a list which represents the successive extension of the field. |
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In the result of @code{sp_noalg} all the algebraic numbers are replaced |
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by the special indeterminate associated with it, that is @code{t#i} |
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for @code{#i}. By this operation the result of @code{sp_noalg} |
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is a list containing only integral polynomials. |
| \E |
\E |
| @item |
@item |
| \BJP |
\BJP |
| Line 1382 the builtin function @code{res()} is always used. |
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| Line 1445 the builtin function @code{res()} is always used. |
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| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item Reference |
\EG @item Reference |
| @fref{asq af}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}. |
@fref{asq af af_noalg}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}. |
| @end table |
@end table |
| |
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