===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/algnum.texi,v
retrieving revision 1.3
retrieving revision 1.7
diff -u -p -r1.3 -r1.7
--- OpenXM/src/asir-doc/parts/algnum.texi	2000/03/10 07:18:40	1.3
+++ OpenXM/src/asir-doc/parts/algnum.texi	2003/04/20 08:01:24	1.7
@@ -1,4 +1,4 @@
-@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 1.6 2003/04/19 15:44:55 noro Exp $
 \BJP
 @node 代数的数に関する演算,,, Top
 @chapter 代数的数に関する演算
@@ -410,13 +410,15 @@ 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
-+((1)/(#0+2))*t#1+((1)/(#0+2))
+(((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1
++((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
-+24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x+(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))
+(((#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+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);
@@ -499,8 +501,8 @@ 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
-+13*A^8-220*A^5-581*A^2)$
+[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+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)
@@ -531,9 +533,9 @@ 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
-+(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3+(#4-23)*x^2+(-21*#4-7)*x
-+(3*#4+8)
+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+(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3
++(#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
@@ -641,9 +643,9 @@ 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
-+(-#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],
-[(#0),t#0^5-2]]]
+[[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),
+2*x+(-#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],[(#0),t#0^5-2]]]
 @end example
 
 @noindent
@@ -706,8 +708,8 @@ may yield a polynomial which differs by a constant.
 * rattoalgp::
 * cr_gcda::
 * sp_norm::
-* asq af::
-* sp::
+* asq af af_noalg::
+* sp sp_noalg::
 @end menu
 
 \JP @node newalg,,, 代数的数に関する函数のまとめ
@@ -1083,7 +1085,7 @@ substitutes a @b{root} for the associated indeterminat
 @item return
 \JP 多項式
 \EG polynomial
-@item poly1, poly2
+@item poly1  poly2
 \JP 多項式
 \EG polynomial
 @end table
@@ -1109,7 +1111,7 @@ x+(-#0)
 @table @t
 \JP @item 参照
 \EG @item Reference
-@fref{gr hgr gr_mod}, @fref{asq af}
+@fref{gr hgr gr_mod}, @fref{asq af af_noalg}
 @end table
 
 \JP @node sp_norm,,, 代数的数に関する函数のまとめ
@@ -1192,14 +1194,15 @@ x^12+2*x^8+5*x^4+1
 @table @t
 \JP @item 参照
 \EG @item Reference
-@fref{res}, @fref{asq af}
+@fref{res}, @fref{asq af af_noalg}
 @end table
 
-\JP @node asq af,,, 代数的数に関する函数のまとめ
-\EG @node asq af,,, Summary of functions for algebraic numbers
-@subsection @code{asq}, @code{af}
+\JP @node asq af af_noalg,,, 代数的数に関する函数のまとめ
+\EG @node asq af af_noalg,,, Summary of functions for algebraic numbers
+@subsection @code{asq}, @code{af}, @code{af_noalg}
 @findex asq
 @findex af
+@findex af_noalg
 
 @table @t
 @item asq(@var{poly})
@@ -1209,6 +1212,7 @@ x^12+2*x^8+5*x^4+1
 algebraic number field.
 \E
 @item af(@var{poly},@var{alglist})
+@itemx af_noalg(@var{poly},@var{defpolylist})
 \JP :: 代数体上の 1 変数多項式の因数分解
 \BEG
 :: Factorization of polynomial @var{poly} over an
@@ -1226,6 +1230,9 @@ algebraic number field.
 @item alglist
 \JP @code{root} のリスト
 \EG @code{root} list
+@item defpolylist
+\JP @code{root} を表す不定元と定義多項式のペアのリスト
+\EG @code{root} list of pairs of an indeterminate and a polynomial
 @end table
 
 @itemize @bullet
@@ -1263,10 +1270,57 @@ In the second argument @code{alglist}, @b{root} define
 first.
 \E
 @item
-\JP 結果は, 通常の無平方分解, 因数分解と同様 [@b{因子}, @b{重複度}] のリストである. 
+\BJP
+@code{af(F,AL)} において, @code{AL} は代数的数のリストであり, 有理数体の
+代数拡大を表す. @code{AL=[An,...,A1]} と書くとき, 各 @code{Ak} は, それより
+右にある代数的数を係数とした, モニックな定義多項式で定義されていなければ
+ならない.
+\E
 \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)}.
+\E       
+
+@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} では, @var{poly} に含まれる代数的数 @var{ai} を不定元 @var{vi}
+で置き換える. @code{defpolylist} は, [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]
+なるリストである. ここで @var{di}(vi,...,v1) は @var{ai} の定義多項式において
+代数的数を全て @var{vj} に置き換えたものである. 
+\E
+\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 [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]. In this expression
+@var{di}(vi,...,v1) is a defining polynomial of @var{ai} represented
+as a multivariate polynomial.
+\E
+
+@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
+結果は, 通常の無平方分解, 因数分解と同様 [@b{因子}, @b{重複度}] 
+のリストである. @code{af_noalg} の場合, @b{因子} に現れる代数的数は, 
+@var{defpolylist} に従って不定元に置き換えられる. 
+\E
+\BEG
 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 
+replaced by the indeterminates according to @var{defpolylist}.
 \E
 @item
 \JP 重複度を込めた因子の全ての積は, @var{poly} と定数倍の違いがあり得る. 
@@ -1277,10 +1331,14 @@ 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
@@ -1289,13 +1347,14 @@ the input polynomial by a constant.
 @fref{cr_gcda}, @fref{fctr sqfr}
 @end table
 
-\JP @node sp,,, 代数的数に関する函数のまとめ
-\EG @node sp,,, Summary of functions for algebraic numbers
-@subsection @code{sp}
+\JP @node sp sp_noalg,,, 代数的数に関する函数のまとめ
+\EG @node sp sp_noalg,,, Summary of functions for algebraic numbers
+@subsection @code{sp}, @code{sp_noalg}
 @findex sp
 
 @table @t
 @item sp(@var{poly})
+@itemx sp_noalg(@var{poly})
 \JP :: 最小分解体を求める. 
 \EG :: Finds the splitting field of polynomial @var{poly} and splits.
 @end table
@@ -1326,12 +1385,18 @@ over the field.
 @item
 \BJP
 結果は, @var{poly} の因子のリストと, 最小分解体の, 逐次拡大による表現
-からなるリストである. 
+からなるリストである. @code{sp_noalg} では, 全ての代数的数が, 対応する
+不定元 (即ち @code{#i} に対する @code{t#i}) に置き換えられる. これに
+より, @code{sp_noalg} の出力は, 整数係数多変数多項式のリストとなる. 
 \E
 \BEG
 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
 a list which represents the successive extension of the field.
+In the result of @code{sp_noalg} all the algebraic numbers are replaced
+by the special indeterminate associated with it, that is @code{t#i}
+for @code{#i}. By this operation the result of @code{sp_noalg}
+is a list containing only integral polynomials.
 \E
 @item
 \BJP
@@ -1342,7 +1407,7 @@ a list which represents the successive extension of th
 \E
 \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
@@ -1369,9 +1434,10 @@ 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),
--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+(-#1^7*#2^3),x+(-#1)],[[(#2),t#2^6+t#1^3*t#2^3+t#1^6],[(#1),t#1^9-54]]]
+[[x+(-#2),-54*x+(#1^6*#2^4),54*x+(#1^6*#2^4+54*#2),
+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*#2^3-54*#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
@@ -1382,6 +1448,6 @@ the builtin function @code{res()} is always used.
 @table @t
 \JP @item 参照
 \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