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Diff for /OpenXM/src/asir-doc/parts/groebner.texi between version 1.2 and 1.4

version 1.2, 1999/12/21 02:47:31 version 1.4, 2003/04/19 15:44:56
Line 1 
Line 1 
 @comment $OpenXM$  @comment $OpenXM: OpenXM/src/asir-doc/parts/groebner.texi,v 1.3 1999/12/24 04:38:04 noro Exp $
 \BJP  \BJP
 @node $B%0%l%V%J4pDl$N7W;;(B,,, Top  @node $B%0%l%V%J4pDl$N7W;;(B,,, Top
 @chapter $B%0%l%V%J4pDl$N7W;;(B  @chapter $B%0%l%V%J4pDl$N7W;;(B
Line 1239  Refer to the sections for each functions.
Line 1239  Refer to the sections for each functions.
 * katsura hkatsura cyclic hcyclic::  * katsura hkatsura cyclic hcyclic::
 * dp_vtoe dp_etov::  * dp_vtoe dp_etov::
 * lex_hensel_gsl tolex_gsl tolex_gsl_d::  * lex_hensel_gsl tolex_gsl tolex_gsl_d::
   * primadec primedec::
 @end menu  @end menu
   
 \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B  \JP @node gr hgr gr_mod,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
Line 1262  Refer to the sections for each functions.
Line 1263  Refer to the sections for each functions.
 @item return  @item return
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item plist, vlist, procs  @item plist  vlist  procs
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 1371  for communication.
Line 1372  for communication.
 @item return  @item return
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item plist, vlist1, vlist2, procs  @item plist  vlist1  vlist2  procs
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 1585  processes.
Line 1586  processes.
 @item return  @item return
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item plist, vlist1, vlist2, procs  @item plist  vlist1  vlist2  procs
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 1691  processes.
Line 1692  processes.
 @item return  @item return
 \JP $BB?9`<0(B  \JP $BB?9`<0(B
 \EG polynomial  \EG polynomial
 @item plist, vlist  @item plist  vlist
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 1788  for @code{gr_minipoly()}.
Line 1789  for @code{gr_minipoly()}.
 @item return  @item return
 \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B  \JP @code{tolexm()} : $B%j%9%H(B, @code{minipolym()} : $BB?9`<0(B
 \EG @code{tolexm()} : list, @code{minipolym()} : polynomial  \EG @code{tolexm()} : list, @code{minipolym()} : polynomial
 @item plist, vlist1, vlist2  @item plist  vlist1  vlist2
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 1853  z^32+11405*z^31+20868*z^30+21602*z^29+...
Line 1854  z^32+11405*z^31+20868*z^30+21602*z^29+...
 @item return  @item return
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item plist, vlist  @item plist  vlist
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 1965  Actual computation is controlled by various parameters
Line 1966  Actual computation is controlled by various parameters
 @item return  @item return
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item plist, vlist  @item plist  vlist
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 2790  selection strategy of critical pairs in Groebner basis
Line 2791  selection strategy of critical pairs in Groebner basis
 @item return  @item return
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @item dpoly1, dpoly2  @item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @end table  @end table
Line 2833  two polynomials, where coefficient is always set to 1.
Line 2834  two polynomials, where coefficient is always set to 1.
 @item return  @item return
 \JP $B@0?t(B  \JP $B@0?t(B
 \EG integer  \EG integer
 @item dpoly1, dpoly2  @item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @end table  @end table
Line 2888  Used for finding candidate terms at reduction of polyn
Line 2889  Used for finding candidate terms at reduction of polyn
 @item return  @item return
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @item dpoly1, dpoly2  @item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @end table  @end table
Line 3112  values of @code{dp_mag()} for intermediate basis eleme
Line 3113  values of @code{dp_mag()} for intermediate basis eleme
 @item return  @item return
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item dpoly1, dpoly2, dpoly3  @item dpoly1  dpoly2  dpoly3
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @item vlist  @item vlist
Line 3136  values of @code{dp_mag()} for intermediate basis eleme
Line 3137  values of @code{dp_mag()} for intermediate basis eleme
 $B$J$i$J$$(B.  $B$J$i$J$$(B.
 @item  @item
 $B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},  $B0z?t$,@0?t78?t$N;~(B, $B4JLs$O(B, $BJ,?t$,8=$l$J$$$h$&(B, $B@0?t(B @var{a}, @var{b},
 $B9`(B @var{t} $B$K$h$j(B @var{a(dpoly1 + dpoly2)-bt dpoly3} $B$H$7$F7W;;$5$l$k(B.  $B9`(B @var{t} $B$K$h$j(B @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3} $B$H$7$F7W;;$5$l$k(B.
 @item  @item
 $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.  $B7k2L$O(B, @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]} $B$J$k%j%9%H$G$"$k(B.
 \E  \E
Line 3155  the divisibility of the head term of @var{dpoly2} by t
Line 3156  the divisibility of the head term of @var{dpoly2} by t
 When integral coefficients, computation is so carefully performed that  When integral coefficients, computation is so carefully performed that
 no rational operations appear in the reduction procedure.  no rational operations appear in the reduction procedure.
 It is computed for integers @var{a} and @var{b}, and a term @var{t} as:  It is computed for integers @var{a} and @var{b}, and a term @var{t} as:
 @var{a(dpoly1 + dpoly2)-bt dpoly3}.  @var{a}(@var{dpoly1} + @var{dpoly2})-@var{bt} @var{dpoly3}.
 @item  @item
 The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.  The result is a list @code{[@var{a dpoly1},@var{a dpoly2 - bt dpoly3}]}.
 \E  \E
Line 3196  The result is a list @code{[@var{a dpoly1},@var{a dpol
Line 3197  The result is a list @code{[@var{a dpoly1},@var{a dpol
 @item return  @item return
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @item dpoly1, dpoly2  @item dpoly1  dpoly2
 \JP $BJ,;6I=8=B?9`<0(B  \JP $BJ,;6I=8=B?9`<0(B
 \EG distributed polynomial  \EG distributed polynomial
 @item mod  @item mod
Line 3272  as a form of @code{[numerator, denominator]})
Line 3273  as a form of @code{[numerator, denominator]})
 @item poly  @item poly
 \JP $BB?9`<0(B  \JP $BB?9`<0(B
 \EG polynomial  \EG polynomial
 @item plist,vlist  @item plist vlist
 \JP $B%j%9%H(B  \JP $B%j%9%H(B
 \EG list  \EG list
 @item order  @item order
Line 3427  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
Line 3428  u0^6,u0^5,u0^4,u0^3,u0^2,u0,1]
 @table @var  @table @var
 \JP @item return 0 $B$^$?$O(B 1  \JP @item return 0 $B$^$?$O(B 1
 \EG @item return 0 or 1  \EG @item return 0 or 1
 @item plist1, plist2  @item plist1  plist2
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
Line 3547  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
Line 3548  u0^2-u0+2*u4^2+2*u3^2+2*u2^2+2*u1^2+2*u5^2]
 @fref{dp_dtop}.  @fref{dp_dtop}.
 @end table  @end table
   
   \JP @node primadec primedec,,, $B%0%l%V%J4pDl$K4X$9$kH!?t(B
   \EG @node primadec primedec,,, Functions for Groebner basis computation
   @subsection @code{primadec}, @code{primedec}
   @findex primadec
   @findex primedec
   
   @table @t
   @item primadec(@var{plist},@var{vlist})
   @item primedec(@var{plist},@var{vlist})
   \JP :: $B%$%G%"%k$NJ,2r(B
   \EG :: Computes decompositions of ideals.
   @end table
   
   @table @var
   @item return
   @itemx plist
   \JP $BB?9`<0%j%9%H(B
   \EG list of polynomials
   @item vlist
   \JP $BJQ?t%j%9%H(B
   \EG list of variables
   @end table
   
   @itemize @bullet
   \BJP
   @item
   @code{primadec()}, @code{primedec} $B$O(B @samp{primdec} $B$GDj5A$5$l$F$$$k(B.
   @item
   @code{primadec()}, @code{primedec()} $B$O$=$l$>$lM-M}?tBN>e$G$N%$%G%"%k$N(B
   $B=`AGJ,2r(B, $B:,4p$NAG%$%G%"%kJ,2r$r9T$&(B.
   @item
   $B0z?t$OB?9`<0%j%9%H$*$h$SJQ?t%j%9%H$G$"$k(B. $BB?9`<0$OM-M}?t78?t$N$_$,5v$5$l$k(B.
   @item
   @code{primadec} $B$O(B @code{[$B=`AG@.J,(B, $BIUB0AG%$%G%"%k(B]} $B$N%j%9%H$rJV$9(B.
   @item
   @code{primadec} $B$O(B $BAG0x;R$N%j%9%H$rJV$9(B.
   @item
   $B7k2L$K$*$$$F(B, $BB?9`<0%j%9%H$H$7$FI=<($5$l$F$$$k3F%$%G%"%k$OA4$F(B
   $B%0%l%V%J4pDl$G$"$k(B. $BBP1~$9$k9`=g=x$O(B, $B$=$l$>$l(B
   $BJQ?t(B @code{PRIMAORD}, @code{PRIMEORD} $B$K3JG<$5$l$F$$$k(B.
   @item
   @code{primadec} $B$O(B @code{[Shimoyama,Yokoyama]} $B$N=`AGJ,2r%"%k%4%j%:%`(B
   $B$r<BAu$7$F$$$k(B.
   @item
   $B$b$7AG0x;R$N$_$r5a$a$?$$$J$i(B, @code{primedec} $B$r;H$&J}$,$h$$(B.
   $B$3$l$O(B, $BF~NO%$%G%"%k$,:,4p%$%G%"%k$G$J$$>l9g$K(B, @code{primadec}
   $B$N7W;;$KM>J,$J%3%9%H$,I,MW$H$J$k>l9g$,$"$k$+$i$G$"$k(B.
   \E
   \BEG
   @item
   Function @code{primadec()} and @code{primedec} are defined in @samp{primdec}.
   @item
   @code{primadec()}, @code{primedec()} are the function for primary
   ideal decomposition and prime decomposition of the radical over the
   rationals respectively.
   @item
   The arguments are a list of polynomials and a list of variables.
   These functions accept ideals with rational function coefficients only.
   @item
   @code{primadec} returns the list of pair lists consisting a primary component
   and its associated prime.
   @item
   @code{primedec} returns the list of prime components.
   @item
   Each component is a Groebner basis and the corresponding term order
   is indicated by the global variables @code{PRIMAORD}, @code{PRIMEORD}
   respectively.
   @item
   @code{primadec} implements the primary decompostion algorithm
   in @code{[Shimoyama,Yokoyama]}.
   @item
   If one only wants to know the prime components of an ideal, then
   use @code{primedec} because @code{primadec} may need additional costs
   if an input ideal is not radical.
   \E
   @end itemize
   
   @example
   [84] load("primdec")$
   [102] primedec([p*q*x-q^2*y^2+q^2*y,-p^2*x^2+p^2*x+p*q*y,
   (q^3*y^4-2*q^3*y^3+q^3*y^2)*x-q^3*y^4+q^3*y^3,
   -q^3*y^4+2*q^3*y^3+(-q^3+p*q^2)*y^2],[p,q,x,y]);
   [[y,x],[y,p],[x,q],[q,p],[x-1,q],[y-1,p],[(y-1)*x-y,q*y^2-2*q*y-p+q]]
   [103] primadec([x,z*y,w*y^2,w^2*y-z^3,y^3],[x,y,z,w]);
   [[[x,z*y,y^2,w^2*y-z^3],[z,y,x]],[[w,x,z*y,z^3,y^3],[w,z,y,x]]]
   @end example
   
   @table @t
   \JP @item $B;2>H(B
   \EG @item References
   @fref{fctr sqfr},
   \JP @fref{$B9`=g=x$N@_Dj(B}.
   \EG @fref{Setting term orderings}.
   @end table
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