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version 1.3, 2000/03/28 02:02:29 version 1.4, 2001/02/27 08:07:24
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 %$OpenXM$  %$OpenXM: OpenXM/doc/compalg/bib.tex,v 1.3 2000/03/28 02:02:29 noro Exp $
 \begin{thebibliography}{99}  \begin{thebibliography}{99}
 \bibitem{ABBOTT}  \bibitem{ABBOTT}
 Abbott, J.A. et al, Factorisation of Polynomials: Old Ideas and Recent Results.  Abbott, J.A. et al, Factorisation of Polynomials: Old Ideas and Recent Results.
Line 89  Giovini, A., Mora, T., Nielsi, G., Robbiano, L., Trave
Line 89  Giovini, A., Mora, T., Nielsi, G., Robbiano, L., Trave
 sugar cube, please'' OR Selection strategies in the Buchberger  sugar cube, please'' OR Selection strategies in the Buchberger
 algorithm.  Proc. ISSAC '91, 49-54.  algorithm.  Proc. ISSAC '91, 49-54.
   
   \bibitem{HOEIJ}
   van Hoeij, M., Factoring polynomials and the knapsack problem.
   To appear in Journal of Number Theory. The preprint is available
   from {\tt http://euclid.math.fsu.edu/\verb+~+hoeij/papers.html}.
   
 \bibitem{KR}  \bibitem{KR}
 $B%+!<%K%O%s(B, B.W., $B%j%C%A!<(B, D.M., $B%W%m%0%i%_%s%08@8l(B C $BBh(B 2 $BHG(B.  $B%+!<%K%O%s(B, B.W., $B%j%C%A!<(B, D.M., $B%W%m%0%i%_%s%08@8l(B C $BBh(B 2 $BHG(B.
 $B6&N)=PHG(B (1989).  $B6&N)=PHG(B (1989).
Line 97  algorithm.  Proc. ISSAC '91, 49-54.
Line 102  algorithm.  Proc. ISSAC '91, 49-54.
 Knuth, D.E., The Art of Computer Programming, Vol. 2.  Knuth, D.E., The Art of Computer Programming, Vol. 2.
 Seminumerical Algorithms, 2nd ed. Addison-Wesley (1981).  Seminumerical Algorithms, 2nd ed. Addison-Wesley (1981).
   
   \bibitem{LENSTRA}
   Lenstra, A.K., Lenstra, H.W., Lob\'asz, Factoring polynomials with
   rational coefficients, Math, Ann. 261 (1982), 515-534.
   
 \bibitem{SUB}  \bibitem{SUB}
 Loos, R., Generalized Polynomial Remainder Sequences.  Loos, R., Generalized Polynomial Remainder Sequences.
 Computing, Suppl. 4 (1982), 115-137.  Computing, Suppl. 4 (1982), 115-137.
Line 261  GCD $B%"%k%4%j%:%`$H$h$P$l$k%?%$%W$N%"%k%4%j%:%`$,$"$
Line 270  GCD $B%"%k%4%j%:%`$H$h$P$l$k%?%$%W$N%"%k%4%j%:%`$,$"$
 $B2r$b(B, $B?t3X$K$*$1$k<B83%D!<%k$H$7$F7W;;5!Be?t%7%9%F%`$rMQ$$$k>l9g$KBgJQ(B  $B2r$b(B, $B?t3X$K$*$1$k<B83%D!<%k$H$7$F7W;;5!Be?t%7%9%F%`$rMQ$$$k>l9g$KBgJQ(B
 $BM-8z$J5!G=$G$"$k(B. $B$3$N>O$NFbMF$N$&$A(B, $BM-8BBN(B, $BM-M}?tBN>e$N0lJQ?tB?9`<0(B  $BM-8z$J5!G=$G$"$k(B. $B$3$N>O$NFbMF$N$&$A(B, $BM-8BBN(B, $BM-M}?tBN>e$N0lJQ?tB?9`<0(B
 $B$N0x?tJ,2r$K4X$7$F$O(B \cite{KNUTH}, $BB?JQ?tB?9`<0$N0x?tJ,2r$K4X$7$F$O(B  $B$N0x?tJ,2r$K4X$7$F$O(B \cite{KNUTH}, $BB?JQ?tB?9`<0$N0x?tJ,2r$K4X$7$F$O(B
 \cite{SASAKI} $B$"$k$$$O(B \cite{GEDDES} $B$r;2>H(B.  \cite{SASAKI} $B$"$k$$$O(B \cite{GEDDES} $B$r;2>H(B. $B$J$*(B, $B%"%k%4%j%:%`(B
   \ref{zassenhaus} $B$O(B $BJ,2r$5$l$kB?9`<0$N<!?t(B $n$ $B$K4X$7$F:G0-7W;;NL$,(B
   $O(2^n)$ $B$H$J$k$,(B, LLL $B%"%k%4%j%:%`(B $B$rMQ$$$kB?9`<0;~4V%"%k%4%j%:%`$,(B
   \cite{LENSTRA} $B$K$h$jDs0F$5$l$F$$$k(B. $B$?$@$7$3$N%"%k%4%j%:%`$O<BMQE*$H(B
   $B$O$$$($:(B, $B0lHL$K$O%"%k%4%j%:%`(B \ref{zassenhaus}$B$,MQ$$$i$l$F$-$?(B. $B$4$/(B
   $B:G6a(B, $B0[$J$k4QE@$+$i(B LLL $B%"%k%4%j%:%`$rMQ$$$kJ}K!$,(B \cite{HOEIJ}
   $B$K$h$jDs0F$5$l$?(B. $B$3$NJ}K!$O(B, $B$3$l$^$G$NJ}K!$G$O;v<B>eJ,2r$,IT2DG=$@$C(B
   $B$?B?9`<0$r8zN($h$/J,2r$9$k$J$I(B, $B<BMQE*$K$bM%$l$F$$$k$3$H$,Js9p$5$l$F$$(B
   $B$k(B.
   
 \noi  \noi
 $BBh(B 7 $B>O(B, $BBh(B 8 $B>O(B:  $BBh(B 7 $B>O(B, $BBh(B 8 $B>O(B:
Line 284  GCD $B%"%k%4%j%:%`$H$h$P$l$k%?%$%W$N%"%k%4%j%:%`$,$"$
Line 301  GCD $B%"%k%4%j%:%`$H$h$P$l$k%?%$%W$N%"%k%4%j%:%`$,$"$
 $BBh(B 10 $B>O(B, $BBh(B 11 $B>O(B:  $BBh(B 10 $B>O(B, $BBh(B 11 $B>O(B:
   
 $B$3$l$i$N>O$O$d$dFC<l$JFbMF$r07$C$F$*$j(B, $B%0%l%V%J4pDl7W;;$r%D!<%k$H$7$F(B  $B$3$l$i$N>O$O$d$dFC<l$JFbMF$r07$C$F$*$j(B, $B%0%l%V%J4pDl7W;;$r%D!<%k$H$7$F(B
 $BMQ$$$k>l9g$K$OFC$K0U<1$9$kI,MW$O$J$$(B. $B$7$+$7(B, $B$=$l$rC1$J$kJXMx$J%V%i%C%/(B  $BMQ$$$k>l9g$K$OFC$K0U<1$9$kI,MW$O$J$$(B. $B$7$+$7(B, $B$=$l$rC1$J$kJXMx$J%V%i%C(B
 $B%\%C%/%9$H9M$($k$H(B, $B$A$g$C$H$7$?LdBj$G$b$9$0$K7W;;$,GKC>$7$F$7$^$&$3$H(B  $B%/%\%C%/%9$H9M$($k$H(B, $B$A$g$C$H$7$?LdBj$G$b$9$0$K7W;;$,GKC>$7$F$7$^$&$3(B
 $B$OCN$C$F$*$/I,MW$,$"$k(B. $BG$0UF~NO$+$i$N%0%l%V%J4pDl7W;;$K$D$$$F$O(B,  $B$H$OCN$C$F$*$/I,MW$,$"$k(B. $BG$0UF~NO$+$i$N%0%l%V%J4pDl7W;;$K$D$$$F$O(B, $B:G(B
 $B:G6aDs0F$5$l$?(B $F_4$ $B%"%k%4%j%:%`(B \cite{F} $B$,M-NO$G$"$k$,(B, $B$^$@(B  $B6aDs0F$5$l$?(B $F_4$ $B%"%k%4%j%:%`(B \cite{F} $B$,M-NO$G$"$k$,(B, $B$^$@HFMQ7W;;(B
 $BHFMQ7W;;5!Be?t%7%9%F%`$J$I$K$O<BAu$5$l$F$$$J$$$h$&$G$"$k(B.  $B5!Be?t%7%9%F%`$J$I$K$O<BAu$5$l$F$$$J$$$h$&$G$"$k(B. $B$^$?(B, change of
 $B$^$?(B, change of ordering $B$K$D$$$F$O(B, $B$3$3$G=R$Y$?J}K!$NB>$K(B  ordering $B$K$D$$$F$O(B, $B$3$3$G=R$Y$?J}K!$NB>$K(B Gr\"obner walk $B$H8F$P$l$kJ}K!$,(B
 \cite{WALK} $B$G0[$J$kJ}K!$,Ds0F$5$l$F$$$k(B. modular change of ordering  \cite{WALK} $B$GDs0F$5$l$F$$$k(B. modular change of ordering$B$*$h$S(B modular
 $B$*$h$S(B modular RUR $B$O(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.  RUR $B$O(B Risa/Asir $B$K<BAu$5$l$F$$$k(B.
   
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