Annotation of OpenXM/src/asir-doc/parts/builtin/poly.texi, Revision 1.1
1.1 ! noro 1: @node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
! 2: @section $BB?9`<0(B, $BM-M}<0$N1i;;(B
! 3:
! 4: @menu
! 5: * var::
! 6: * vars::
! 7: * uc::
! 8: * coef::
! 9: * deg mindeg::
! 10: * nmono::
! 11: * ord::
! 12: * sdiv sdivm srem sremm sqr sqrm::
! 13: * tdiv::
! 14: * %::
! 15: * subst psubst::
! 16: * diff::
! 17: * res::
! 18: * fctr sqfr::
! 19: * modfctr::
! 20: * ufctrhint::
! 21: * ptozp::
! 22: * prim cont::
! 23: * gcd gcdz::
! 24: * red::
! 25: @end menu
! 26:
! 27: @node var,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 28: @subsection @code{var}
! 29: @findex var
! 30:
! 31: @table @t
! 32: @item var(@var{rat})
! 33: :: @var{rat} $B$N<gJQ?t(B.
! 34: @end table
! 35:
! 36: @table @var
! 37: @item return
! 38: $BITDj85(B
! 39: @item rat
! 40: $BM-M}<0(B
! 41: @end table
! 42:
! 43: @itemize @bullet
! 44: @item
! 45: $B<gJQ?t$K4X$7$F$O(B, @xref{Asir $B$G;HMQ2DG=$J7?(B}.
! 46: @item
! 47: $B%G%U%)%k%H$NJQ?t=g=x$O<!$N$h$&$K$J$C$F$$$k(B.
! 48:
! 49: @code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q}, @code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e},
! 50: @code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m}, @code{n}, @code{o},$B0J8e$OJQ?t$N8=$l$?=g(B.
! 51: @end itemize
! 52:
! 53: @example
! 54: [0] var(x^2+y^2+a^2);
! 55: x
! 56: [1] var(a*b*c*d*e);
! 57: a
! 58: [2] var(3/abc+2*xy/efg);
! 59: abc
! 60: @end example
! 61:
! 62: @table @t
! 63: @item $B;2>H(B
! 64: @fref{ord}, @fref{vars}.
! 65: @end table
! 66:
! 67: @node vars,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 68: @subsection @code{vars}
! 69: @findex vars
! 70:
! 71: @table @t
! 72: @item vars(@var{obj})
! 73: :: @var{obj} $B$K4^$^$l$kJQ?t$N%j%9%H(B.
! 74: @end table
! 75:
! 76: @table @var
! 77: @item return
! 78: $B%j%9%H(B
! 79: @item obj
! 80: $BG$0U(B
! 81: @end table
! 82:
! 83: @itemize @bullet
! 84: @item
! 85: $BM?$($i$l$?<0$K4^$^$l$kJQ?t$N%j%9%H$rJV$9(B.
! 86: @item
! 87: $BJQ?t=g=x$N9b$$$b$N$+$i=g$KJB$Y$k(B.
! 88: @end itemize
! 89:
! 90: @example
! 91: [0] vars(x^2+y^2+a^2);
! 92: [x,y,a]
! 93: [1] vars(3/abc+2*xy/efg);
! 94: [abc,xy,efg]
! 95: [2] vars([x,y,z]);
! 96: [x,y,z]
! 97: @end example
! 98:
! 99: @table @t
! 100: @item $B;2>H(B
! 101: @fref{var}, @fref{uc}, @fref{ord}.
! 102: @end table
! 103:
! 104: @node uc,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 105: @subsection @code{uc}
! 106: @findex uc
! 107:
! 108: @table @t
! 109: @item uc()
! 110: :: $B?7$?$JITDj85$r@8@.$9$k(B.
! 111: @end table
! 112:
! 113: @table @var
! 114: @item return
! 115: @code{vtype} $B$,(B 1 $B$NITDj85(B
! 116: @end table
! 117:
! 118: @itemize @bullet
! 119: @item
! 120: @code{uc()} $B$r<B9T$9$k$?$S$K(B, @code{_0}, @code{_1}, @code{_2},... $B$H$$$&(B
! 121: $BITDj85$r@8@.$9$k(B.
! 122: @item
! 123: @code{uc()} $B$G@8@.$5$l$?ITDj85$O(B, $BD>@\%-!<%\!<%I$+$iF~NO$9$k$3$H$,$G$-$J$$(B.
! 124: $B$3$l$O(B, $B%W%m%0%i%`Cf$GL$Dj78?t$r<+F0@8@.$9$k>l9g(B, $BF~NO$J$I$K4^$^$l$k(B
! 125: $BITDj85$HF10l$N$b$N$,@8@.$5$l$k$3$H$rKI$0$?$a$G$"$k(B.
! 126: @item
! 127: $BDL>o$NITDj85(B (@code{vtype} $B$,(B 0) $B$N<+F0@8@.$K$O(B @code{rtostr()},
! 128: @code{strtov()} $B$rMQ$$$k(B.
! 129: @item
! 130: @code{uc()} $B$G@8@.$5$l$?ITDj85$NITDj85$H$7$F$N7?(B (@code{vtype}) $B$O(B 1 $B$G$"$k(B.
! 131: (@xref{$BITDj85$N7?(B})
! 132: @end itemize
! 133:
! 134: @example
! 135: [0] A=uc();
! 136: _0
! 137: [1] B=uc();
! 138: _1
! 139: [2] (uc()+uc())^2;
! 140: _2^2+2*_3*_2+_3^2
! 141: [3] (A+B)^2;
! 142: _0^2+2*_1*_0+_1^2
! 143: @end example
! 144:
! 145: @table @t
! 146: @item $B;2>H(B
! 147: @fref{vtype}, @fref{rtostr}, @fref{strtov}.
! 148: @end table
! 149:
! 150: @node coef,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 151: @subsection @code{coef}
! 152: @findex coef
! 153:
! 154: @table @t
! 155: @item coef(@var{poly},@var{deg}[,@var{var}])
! 156: :: @var{poly} $B$N(B @var{var} ($B>JN,;~$O<gJQ?t(B) $B$K4X$9$k(B @var{deg} $B<!$N78?t(B.
! 157: @end table
! 158:
! 159: @table @var
! 160: @item return
! 161: $BB?9`<0(B
! 162: @item poly
! 163: $BB?9`<0(B
! 164: @item var
! 165: $BITDj85(B
! 166: @item deg
! 167: $B<+A3?t(B
! 168: @end table
! 169:
! 170: @itemize @bullet
! 171: @item
! 172: @var{poly} $B$N(B @var{var} $B$K4X$9$k(B @var{deg} $B<!$N78?t$r=PNO$9$k(B.
! 173: @item
! 174: @var{var} $B$O(B, $B>JN,$9$k$H<gJQ?t(B @t{var}(@var{poly}) $B$@$H$_$J$5$l$k(B.
! 175: @item
! 176: @var{var} $B$,<gJQ?t$G$J$$;~(B, @var{var} $B$,<gJQ?t$N>l9g$KHf3S$7$F(B
! 177: $B8zN($,Mn$A$k(B.
! 178: @end itemize
! 179:
! 180: @example
! 181: [0] A = (x+y+z)^3;
! 182: x^3+(3*y+3*z)*x^2+(3*y^2+6*z*y+3*z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
! 183: [1] coef(A,1,y);
! 184: 3*x^2+6*z*x+3*z^2
! 185: [2] coef(A,0);
! 186: y^3+3*z*y^2+3*z^2*y+z^3
! 187: @end example
! 188:
! 189: @table @t
! 190: @item $B;2>H(B
! 191: @fref{var}, @fref{deg mindeg}.
! 192: @end table
! 193:
! 194: @node deg mindeg,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 195: @subsection @code{deg}, @code{mindeg}
! 196: @findex deg
! 197: @findex mindeg
! 198:
! 199: @table @t
! 200: @item deg(@var{poly},@var{var})
! 201: :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B.
! 202: @item mindeg(@var{poly},@var{var})
! 203: :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B.
! 204: @end table
! 205:
! 206: @table @var
! 207: @item return
! 208: $B<+A3?t(B
! 209: @item poly
! 210: $BB?9`<0(B
! 211: @item var
! 212: $BITDj85(B
! 213: @end table
! 214:
! 215: @itemize @bullet
! 216: @item
! 217: $BM?$($i$l$?B?9`<0$NJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B, $B:GDc<!?t$r=PNO$9$k(B.
! 218: @item
! 219: $BJQ?t(B @var{var} $B$r>JN,$9$k$3$H$O=PMh$J$$(B.
! 220: @end itemize
! 221:
! 222: @example
! 223: [0] deg((x+y+z)^10,x);
! 224: 10
! 225: [1] deg((x+y+z)^10,w);
! 226: 0
! 227: [75] mindeg(x^2+3*x*y,x);
! 228: 1
! 229: @end example
! 230:
! 231: @node nmono,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 232: @subsection @code{nmono}
! 233: @findex nmono
! 234:
! 235: @table @t
! 236: @item nmono(@var{rat})
! 237: :: @var{rat} $B$NC19`<0$N9`?t(B.
! 238: @end table
! 239:
! 240: @table @var
! 241: @item return
! 242: $B<+A3?t(B
! 243: @item rat
! 244: $BM-M}<0(B
! 245: @end table
! 246:
! 247: @itemize @bullet
! 248: @item
! 249: $BB?9`<0$rE83+$7$?>uBV$G$N(B 0 $B$G$J$$78?t$r;}$DC19`<0$N9`?t$r5a$a$k(B.
! 250: @item
! 251: $BM-M}<0$N>l9g$O(B, $BJ,;R$HJ,Jl$N9`?t$NOB$,JV$5$l$k(B.
! 252: @item
! 253: $BH!?t7A<0(B (@xref{$BITDj85$N7?(B}) $B$O(B, $B0z?t$,2?$G$"$C$F$bC19`$H$_$J$5$l$k(B. (1 $B8D$NITDj85$HF1$8(B. )
! 254: @end itemize
! 255:
! 256: @example
! 257: [0] nmono((x+y)^10);
! 258: 11
! 259: [1] nmono((x+y)^10/(x+z)^10);
! 260: 22
! 261: [2] nmono(sin((x+y)^10));
! 262: 1
! 263: @end example
! 264:
! 265: @table @t
! 266: @item $B;2>H(B
! 267: @fref{vtype}.
! 268: @end table
! 269:
! 270: @node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 271: @subsection @code{ord}
! 272: @findex ord
! 273:
! 274: @table @t
! 275: @item ord([@var{varlist}])
! 276: :: $BJQ?t=g=x$N@_Dj(B
! 277: @end table
! 278:
! 279: @table @var
! 280: @item return
! 281: $BJQ?t$N%j%9%H(B
! 282: @item varlist
! 283: $BJQ?t$N%j%9%H(B
! 284: @end table
! 285:
! 286: @itemize @bullet
! 287: @item
! 288: $B0z?t$,$"$k$H$-(B, $B0z?t$NJQ?t%j%9%H$r@hF,$K=P$7(B, $B;D$j$NJQ?t$,$=$N8e$K(B
! 289: $BB3$/$h$&$KJQ?t=g=x$r@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K4X$o$i$:(B, @code{ord()}
! 290: $B$N=*N;;~$K$*$1$kJQ?t=g=x%j%9%H$rJV$9(B.
! 291:
! 292: @item
! 293: $B$3$NH!?t$K$h$kJQ?t=g=x$NJQ99$r9T$C$F$b(B, $B4{$K%W%m%0%i%`JQ?t$J$I$K(B
! 294: $BBeF~$5$l$F$$$k<0$NFbIt7A<0$O?7$7$$=g=x$K=>$C$F$OJQ99$5$l$J$$(B.
! 295: $B=>$C$F(B, $B$3$NH!?t$K$h$k=g=x$NJQ99$O(B, @b{Asir} $B$N5/F0D>8e(B,
! 296: $B$"$k$$$O(B, $B?7$?$JJQ?t$,8=$l$?;~E@$K9T$o$l$k(B
! 297: $B$Y$-$G$"$k(B. $B0[$J$kJQ?t=g=x$N$b$H$G@8@.$5$l$?<0$I$&$7$N1i;;(B
! 298: $B$,9T$o$l$?>l9g(B, $BM=4|$;$L7k2L$,@8$:$k$3$H$b$"$jF@$k(B.
! 299: @end itemize
! 300:
! 301: @example
! 302: [0] ord();
! 303: [x,y,z,u,v,w,p,q,r,s,t,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,_w,_p,
! 304: _q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),(_x)^(_y),
! 305: log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),
! 306: tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
! 307: [1] ord([dx,dy,dz,a,b,c]);
! 308: [dx,dy,dz,a,b,c,x,y,z,u,v,w,p,q,r,s,t,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,
! 309: _w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),
! 310: (_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),
! 311: cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
! 312: @end example
! 313:
! 314: @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 315: @subsection @code{sdiv}, @code{sdivm}, @code{srem}, @code{sremm}, @code{sqr}, @code{sqrm}
! 316: @findex sdiv
! 317: @findex sdivm
! 318: @findex srem
! 319: @findex sremm
! 320: @findex sqr
! 321: @findex sqrm
! 322:
! 323: @table @t
! 324: @item sdiv(@var{poly1},@var{poly2}[,@var{v}])
! 325: @itemx sdivm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
! 326: :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&$r5a$a$k(B.
! 327: @item srem(@var{poly1},@var{poly2}[,@var{v}])
! 328: @item sremm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
! 329: :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>jM>$r5a$a$k(B.
! 330: @item sqr(@var{poly1},@var{poly2}[,@var{v}])
! 331: @item sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
! 332: :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&(B, $B>jM>$r(B
! 333: $B5a$a$k(B.
! 334: @end table
! 335:
! 336: @table @var
! 337: @item return
! 338: @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : $BB?9`<0(B, @code{sqr()}, @code{sqrm()} : @code{[$B>&(B,$B>jM>(B]} $B$J$k%j%9%H(B
! 339: @item poly1 poly2
! 340: $BB?9`<0(B
! 341: @item v
! 342: $BITDj85(B
! 343: @item mod
! 344: $BAG?t(B
! 345: @end table
! 346:
! 347: @itemize @bullet
! 348: @item
! 349: @var{poly1} $B$r(B @var{poly2} $B$N<gJQ?t(B @t{var}(@var{poly2})
! 350: ( $B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) $B$K4X$9$kB?9`<0$H8+$F(B,
! 351: @var{poly2} $B$G(B, $B3d$j;;$r9T$&(B.
! 352: @item
! 353: @code{sdivm()}, @code{sremm()}, @code{sqrm()} $B$O(B GF(@var{mod}) $B>e$G7W;;$9$k(B.
! 354: @item
! 355: $BB?9`<0$N=|;;$O(B, $B<g78?t$I$&$7$N3d;;$K$h$jF@$i$l$?>&$H(B, $B<gJQ?t$NE,Ev$JQQ$N(B
! 356: $B@Q$r(B @var{poly2} $B$K3]$1$F(B, @var{poly1} $B$+$i0z$/$H$$$&A`:n$r(B
! 357: @var{poly1} $B$N<!?t$,(B @var{poly2} $B$N<!?t$h$j>.$5$/$J$k$^$G7+$jJV$7$F(B
! 358: $B9T$&(B. $B$3$NA`:n$,(B, $BB?9`<0$NHO0OFb$G9T$o$l$k$?$a$K$O(B, $B3F%9%F%C%W$K$*$$$F(B
! 359: $B<g78?t$I$&$7$N=|;;$,(B, $BB?9`<0$H$7$F$N@0=|$G$"$kI,MW$,$"$k(B. $B$3$l$,(B, $B!V=|;;(B
! 360: $B$,:G8e$^$G<B9T$G$-$k!W$3$H$N0UL#$G$"$k(B.
! 361: @item
! 362: $BE57?E*$J>l9g$H$7$F(B, @var{poly2} $B$N<g78?t$,(B, $BM-M}?t$G$"$k>l9g(B, $B$"$k$$$O(B,
! 363: @var{poly2} $B$,(B @var{poly1} $B$N0x;R$G$"$k$3$H$,$o$+$C$F$$$k>l9g$J$I(B
! 364: $B$,$"$k(B.
! 365: @item
! 366: @code{sqr()} $B$O>&$H>jM>$rF1;~$K5a$a$?$$;~$KMQ$$$k(B.
! 367: @item
! 368: $B@0?t=|;;$N>&(B, $B>jM>$O(B @code{idiv}, @code{irem} $B$rMQ$$$k(B.
! 369: @item
! 370: $B78?t$KBP$9$k>jM>1i;;$O(B @code{%} $B$rMQ$$$k(B.
! 371: @end itemize
! 372:
! 373: @example
! 374: [0] sdiv((x+y+z)^3,x^2+y+a);
! 375: x+3*y+3*z
! 376: [1] srem((x+y+z)^2,x^2+y+a);
! 377: (2*y+2*z)*x+y^2+(2*z-1)*y+z^2-a
! 378: [2] X=(x+y+z)*(x-y-z)^2;
! 379: x^3+(-y-z)*x^2+(-y^2-2*z*y-z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
! 380: [3] Y=(x+y+z)^2*(x-y-z);
! 381: x^3+(y+z)*x^2+(-y^2-2*z*y-z^2)*x-y^3-3*z*y^2-3*z^2*y-z^3
! 382: [4] G=gcd(X,Y);
! 383: x^2-y^2-2*z*y-z^2
! 384: [5] sqr(X,G);
! 385: [x-y-z,0]
! 386: [6] sqr(Y,G);
! 387: [x+y+z,0]
! 388: [7] sdiv(y*x^3+x+1,y*x+1);
! 389: divsp: cannot happen
! 390: return to toplevel
! 391: @end example
! 392:
! 393: @table @t
! 394: @item $B;2>H(B
! 395: @fref{idiv irem}, @fref{%}.
! 396: @end table
! 397:
! 398: @node tdiv,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 399: @subsection @code{tdiv}
! 400: @findex tdiv
! 401:
! 402: @table @t
! 403: @item tdiv(@var{poly1},@var{poly2})
! 404: :: @var{poly1} $B$,(B @var{poly2} $B$G3d$j@Z$l$k$+$I$&$+D4$Y$k(B.
! 405: @end table
! 406:
! 407: @table @var
! 408: @item return
! 409: $B3d$j@Z$l$k$J$i$P>&(B, $B3d$j@Z$l$J$1$l$P(B 0
! 410: @item poly1 poly2
! 411: $BB?9`<0(B
! 412: @end table
! 413:
! 414: @itemize @bullet
! 415: @item
! 416: @var{poly2} $B$,(B @var{poly1} $B$rB?9`<0$H$7$F3d$j@Z$k$+$I$&$+D4$Y$k(B.
! 417: @item
! 418: $B$"$kB?9`<0$,4{Ls0x;R$G$"$k$3$H$O$o$+$C$F$$$k$,(B, $B$=$N=EJ#EY$,$o$+$i$J$$(B
! 419: $B>l9g$K(B, @code{tdiv()} $B$r7+$jJV$78F$V$3$H$K$h$j=EJ#EY$,$o$+$k(B.
! 420: @end itemize
! 421:
! 422: @example
! 423: [11] Y=(x+y+z)^5*(x-y-z)^3;
! 424: x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5
! 425: +(6*y^5+30*z*y^4+60*z^2*y^3+60*z^3*y^2+30*z^4*y+6*z^5)*x^3+(2*y^6+12*z*y^5
! 426: +30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2+(-2*y^7-14*z*y^6
! 427: -42*z^2*y^5-70*z^3*y^4-70*z^4*y^3-42*z^5*y^2-14*z^6*y-2*z^7)*x-y^8-8*z*y^7
! 428: -28*z^2*y^6-56*z^3*y^5-70*z^4*y^4-56*z^5*y^3-28*z^6*y^2-8*z^7*y-z^8
! 429: [12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++);
! 430: [13] I;
! 431: 5
! 432: @end example
! 433:
! 434: @table @t
! 435: @item $B;2>H(B
! 436: @fref{sdiv sdivm srem sremm sqr sqrm}.
! 437: @end table
! 438:
! 439: @node %,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 440: @subsection @code{%}
! 441: @findex %
! 442:
! 443: @table @t
! 444: @item @var{poly} % @var{m}
! 445: :: $B@0?t$K$h$k>jM>(B
! 446: @end table
! 447:
! 448: @table @var
! 449: @item return
! 450: $B@0?t$^$?$OB?9`<0(B
! 451: @item poly
! 452: $B@0?t$^$?$O@0?t78?tB?9`<0(B
! 453: @item m
! 454: $B@0?t(B
! 455: @end table
! 456:
! 457: @itemize @bullet
! 458: @item
! 459: @var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B.
! 460: @item
! 461: $B7k2L$N78?t$OA4$F@5$N@0?t$H$J$k(B.
! 462: @item
! 463: @var{poly} $B$O@0?t$G$b$h$$(B. $B$3$N>l9g(B, $B7k2L$,@5$K@55,2=$5$l$k$3$H$r=|$1$P(B
! 464: @code{irem()} $B$HF1MM$KMQ$$$k$3$H$,$G$-$k(B.
! 465: @item
! 466: @var{poly} $B$N78?t(B, @var{m} $B$H$b@0?t$G$"$kI,MW$,$"$k$,(B, $B%A%'%C%/$O9T$J$o$l$J$$(B.
! 467: @end itemize
! 468:
! 469: @example
! 470: [0] (x+2)^5 % 3;
! 471: x^5+x^4+x^3+2*x^2+2*x+2
! 472: [1] (x-2)^5 % 3;
! 473: x^5+2*x^4+x^3+x^2+2*x+1
! 474: [2] (-5) % 4;
! 475: 3
! 476: [3] irem(-5,4);
! 477: -1
! 478: @end example
! 479:
! 480: @table @t
! 481: @item $B;2>H(B
! 482: @fref{idiv irem}.
! 483: @end table
! 484:
! 485: @node subst psubst,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 486: @subsection @code{subst}, @code{psubst}
! 487: @findex subst
! 488: @findex psubst
! 489:
! 490: @table @t
! 491: @item subst(@var{rat}[,@var{varn},@var{ratn}]*)
! 492: @item psubst(@var{rat}[,@var{var},@var{rat}]*)
! 493: :: @var{rat} $B$N(B @var{varn} $B$K(B @var{ratn} $B$rBeF~(B
! 494: (@var{n=1,2},... $B$G:8$+$i1&$K=g<!BeF~$9$k(B).
! 495: @end table
! 496:
! 497: @table @var
! 498: @item return
! 499: $BM-M}<0(B
! 500: @item rat,ratn
! 501: $BM-M}<0(B
! 502: @item varn
! 503: $BITDj85(B
! 504: @end table
! 505:
! 506: @itemize @bullet
! 507: @item
! 508: $BM-M}<0$NFCDj$NITDj85$K(B, $BDj?t$"$k$$$OB?9`<0(B, $BM-M}<0$J$I$rBeF~$9$k$N$KMQ$$$k(B.
! 509: @item
! 510: @t{subst}(@var{rat},@var{var1},@var{rat1},@var{var2},@var{rat2},...) $B$O(B,
! 511: @t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},...)
! 512: $B$HF1$80UL#$G$"$k(B.
! 513: @item
! 514: $BF~NO$N:8B&$+$i=g$KBeF~$r7+$jJV$9$?$a$K(B, $BF~NO$N=g$K$h$C$F7k2L$,JQ$o$k$3$H$,$"$k(B.
! 515: @item
! 516: @code{subst()} $B$O(B, @code{sin()} $B$J$I$NH!?t$N0z?t$KBP$7$F$bBeF~$r9T$&(B.
! 517: @code{psubst()} $B$O(B, $B$3$N$h$&$JH!?t$r0l$D$NFHN)$7$?ITDj85$H8+$J$7$F(B, $B$=(B
! 518: $B$N0z?t$K$OBeF~$O9T$o$J$$(B. (partial substitution $B$N$D$b$j(B)
! 519: @item
! 520: @b{Asir} $B$G$O(B, $BM-M}<0$NLsJ,$O<+F0E*$K$O9T$o$J$$$?$a(B,
! 521: $BM-M}<0$NBeF~$O(B, $B;W$o$L7W;;;~4V$NA}Bg$r0z$-5/$3$9>l9g$,$"$k(B.
! 522: $BM-M}<0$rBeF~$9$k>l9g$K$O(B, $BLdBj$K1~$8$?FH<+$NH!?t$r=q$$$F(B,
! 523: $B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B.
! 524: @item
! 525: $BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B.
! 526: @end itemize
! 527:
! 528: @example
! 529: [0] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2);
! 530: x^3-6*x^2+12*x-8
! 531: [1] subst(@@@@,x,-1);
! 532: -27
! 533: [2] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2,x,-1);
! 534: -27
! 535: [3] subst(x*y^3,x,y,y,x);
! 536: x^4
! 537: [4] subst(x*y^3,y,x,x,y);
! 538: y^4
! 539: [5] subst(x*y^3,x,t,y,x,t,y);
! 540: y*x^3
! 541: [6] subst(x*sin(x),x,t);
! 542: sint(t)*t
! 543: [7] psubst(x*sin(x),x,t);
! 544: sin(x)*t
! 545: @end example
! 546:
! 547: @node diff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 548: @subsection @code{diff}
! 549: @findex diff
! 550:
! 551: @table @t
! 552: @item diff(@var{rat}[,@var{varn}]*)
! 553: @item diff(@var{rat},@var{varlist})
! 554: :: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!HyJ,$9$k(B.
! 555: @end table
! 556:
! 557: @table @var
! 558: @item return
! 559: $B<0(B
! 560: @item rat
! 561: $BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B)
! 562: @item varn
! 563: $BITDj85(B
! 564: @item varlist
! 565: $BITDj85$N%j%9%H(B
! 566: @end table
! 567:
! 568: @itemize @bullet
! 569: @item
! 570: $BM?$($i$l$?=iEyH!?t$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G(B
! 571: $B=g<!HyJ,$9$k(B.
! 572: @item
! 573: $B:8B&$NITDj85$h$j(B, $B=g$KHyJ,$7$F$$$/(B. $B$D$^$j(B, @t{diff}(@var{rat},@t{x,y}) $B$O(B,
! 574: @t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}) $B$HF1$8$G$"$k(B.
! 575: @end itemize
! 576:
! 577: @example
! 578: [0] diff((x+2*y)^2,x);
! 579: 2*x+4*y
! 580: [1] diff((x+2*y)^2,x,y);
! 581: 4
! 582: [2] diff(x/sin(log(x)+1),x);
! 583: (sin(log(x)+1)-cos(log(x)+1))/(sin(log(x)+1)^2)
! 584: [3] diff(sin(x),[x,x,x,x]);
! 585: sin(x)
! 586: @end example
! 587:
! 588: @node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 589: @subsection @code{res}
! 590: @findex res
! 591:
! 592: @table @t
! 593: @item res(@var{var},@var{poly1},@var{poly2}[,@var{mod}])
! 594: :: @var{var} $B$K4X$9$k(B @var{poly1} $B$H(B @var{poly2} $B$N=*7k<0(B.
! 595: @end table
! 596:
! 597: @table @var
! 598: @item return
! 599: $BB?9`<0(B
! 600: @item var
! 601: $BITDj85(B
! 602: @item poly1,poly2
! 603: $BB?9`<0(B
! 604: @item mod
! 605: $BAG?t(B
! 606: @end table
! 607:
! 608: @itemize @bullet
! 609: @item
! 610: $BFs$D$NB?9`<0(B @var{poly1} $B$H(B @var{poly2} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k(B
! 611: $B=*7k<0$r5a$a$k(B.
! 612: @item
! 613: $BItJ,=*7k<0%"%k%4%j%:%`$K$h$k(B.
! 614: @item
! 615: $B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N7W;;$r9T$&(B.
! 616: @end itemize
! 617:
! 618: @example
! 619: [0] res(t,(t^3+1)*x+1,(t^3+1)*y+t);
! 620: -x^3-x^2-y^3
! 621: @end example
! 622:
! 623: @node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 624: @subsection @code{fctr}, @code{sqfr}
! 625: @findex fctr
! 626: @findex sqfr
! 627:
! 628: @table @t
! 629: @item fctr(@var{poly})
! 630: :: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.
! 631: @item sqfr(@var{poly})
! 632: :: @var{poly} $B$rL5J?J}J,2r$9$k(B.
! 633: @end table
! 634:
! 635: @table @var
! 636: @item return
! 637: $B%j%9%H(B
! 638: @item poly
! 639: $BM-M}?t78?t$NB?9`<0(B
! 640: @end table
! 641:
! 642: @itemize @bullet
! 643: @item
! 644: $BM-M}?t78?t$NB?9`<0(B @var{poly} $B$r0x?tJ,2r$9$k(B. @code{fctr()} $B$O4{Ls0x;RJ,2r(B,
! 645: @code{sqfr()} $B$OL5J?J}0x;RJ,2r(B.
! 646: @item
! 647: $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.
! 648: @item
! 649: @b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B.
! 650: @item
! 651: @b{$B?t78?t(B} $B$O(B, (@var{poly}/@b{$B?t78?t(B}) $B$,(B, $B@0?t78?t$G(B, $B78?t$N(B GCD $B$,(B 1 $B$H$J$k(B
! 652: $B$h$&$JB?9`<0$K$J$k$h$&$KA*$P$l$F$$$k(B. (@code{ptozp()} $B;2>H(B)
! 653: @end itemize
! 654:
! 655: @example
! 656: [0] fctr(x^10-1);
! 657: [[1,1],[x-1,1],[x+1,1],[x^4+x^3+x^2+x+1,1],[x^4-x^3+x^2-x+1,1]]
! 658: [1] fctr(x^3+y^3+(z/3)^3-x*y*z);
! 659: [[1/27,1],[9*x^2+(-9*y-3*z)*x+9*y^2-3*z*y+z^2,1],[3*x+3*y+z,1]]
! 660: [2] A=(a+b+c+d)^2;
! 661: a^2+(2*b+2*c+2*d)*a+b^2+(2*c+2*d)*b+c^2+2*d*c+d^2
! 662: [3] fctr(A);
! 663: [[1,1],[a+b+c+d,2]]
! 664: [4] A=(x+1)*(x^2-y^2)^2;
! 665: x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4
! 666: [5] sqfr(A);
! 667: [[1,1],[x+1,1],[-x^2+y^2,2]]
! 668: [6] fctr(A);
! 669: [[1,1],[x+1,1],[-x-y,2],[x-y,2]]
! 670: @end example
! 671:
! 672: @table @t
! 673: @item $B;2>H(B
! 674: @fref{ufctrhint}.
! 675: @end table
! 676:
! 677: @node ufctrhint,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 678: @subsection @code{ufctrhint}
! 679: @findex ufctrhint
! 680:
! 681: @table @t
! 682: @item ufctrhint(@var{poly},@var{hint})
! 683: :: $B<!?t>pJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
! 684: @end table
! 685:
! 686: @table @var
! 687: @item return
! 688: $B%j%9%H(B
! 689: @item poly
! 690: $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B
! 691: @item hint
! 692: $B<+A3?t(B
! 693: @end table
! 694:
! 695: @itemize @bullet
! 696: @item
! 697: $B3F4{Ls0x;R$N<!?t$,(B @var{hint} $B$NG\?t$G$"$k$3$H$,$o$+$C$F$$$k>l9g$K(B
! 698: @var{poly} $B$N4{Ls0x;RJ,2r$r(B @code{fctr()} $B$h$j8zN(NI$/9T$&(B.
! 699: @var{poly} $B$,(B, @var{d} $B<!$N3HBgBN>e$K$*$1$k(B
! 700: $B$"$kB?9`<0$N%N%k%`(B (@xref{$BBe?tE*?t$K4X$9$k1i;;(B}) $B$GL5J?J}$G$"$k>l9g(B,
! 701: $B3F4{Ls0x;R$N<!?t$O(B @var{d} $B$NG\?t$H$J$k(B. $B$3$N$h$&$J>l9g$K(B
! 702: $BMQ$$$i$l$k(B.
! 703: @end itemize
! 704:
! 705: @example
! 706: [10] A=t^9-15*t^6-87*t^3-125;
! 707: t^9-15*t^6-87*t^3-125
! 708: 0msec
! 709: [11] N=res(t,subst(A,t,x-2*t),A);
! 710: -x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69+1720634125410*x^66
! 711: -88249977024390*x^63-4856095669551930*x^60+1999385245240571421*x^57
! 712: -15579689952590251515*x^54+15956967531741971462865*x^51
! 713: ...
! 714: +140395588720353973535526123612661444550659875*x^6
! 715: +10122324287343155430042768923500799484375*x^3
! 716: +139262743444407310133459021182733314453125
! 717: 980msec + gc : 250msec
! 718: [12] sqfr(N);
! 719: [[-1,1],[x^81-1215*x^78+567405*x^75-139519665*x^72+19360343142*x^69
! 720: -1720634125410*x^66+88249977024390*x^63+4856095669551930*x^60
! 721: -1999385245240571421*x^57+15579689952590251515*x^54
! 722: ...
! 723: -10122324287343155430042768923500799484375*x^3
! 724: -139262743444407310133459021182733314453125,1]]
! 725: 20msec
! 726: [13] fctr(N);
! 727: [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
! 728: [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
! 729: +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
! 730: [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],
! 731: [x^18+10773*x^12+2784051*x^6+307546875,1]]
! 732: 167.050sec + gc : 1.890sec
! 733: [14] ufctrhint(N,9);
! 734: [[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
! 735: [x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
! 736: +296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
! 737: [x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],
! 738: [x^18+10773*x^12+2784051*x^6+307546875,1]]
! 739: 119.340sec + gc : 1.300sec
! 740: @end example
! 741:
! 742: @table @t
! 743: @item $B;2>H(B
! 744: @fref{fctr sqfr}.
! 745: @end table
! 746:
! 747: @node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 748: @subsection @code{modfctr}
! 749: @findex modfctr
! 750:
! 751: @table @t
! 752: @item modfctr(@var{poly},@var{mod})
! 753: :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
! 754: @end table
! 755:
! 756: @table @var
! 757: @item return
! 758: $B%j%9%H(B
! 759: @item poly
! 760: $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B
! 761: @item mod
! 762: $B<+A3?t(B
! 763: @end table
! 764:
! 765: @itemize @bullet
! 766: @item
! 767: 2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B
! 768: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.
! 769: @item
! 770: $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.
! 771: @item
! 772: @b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B.
! 773: @end itemize
! 774:
! 775: @example
! 776: [0] modfctr(x^10+x^2+1,2147483647);
! 777: [[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],
! 778: [x+634005911,1],[x+1513477735,1],[x+634005912,1],
! 779: [x^4+1759639395*x^2+2045307031,1]]
! 780: @end example
! 781:
! 782: @table @t
! 783: @item $B;2>H(B
! 784: @fref{fctr sqfr}.
! 785: @end table
! 786:
! 787: @node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 788: @subsection @code{ptozp}
! 789: @findex ptozp
! 790:
! 791: @table @t
! 792: @item ptozp(@var{poly})
! 793: :: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B.
! 794: @end table
! 795:
! 796: @table @var
! 797: @item return
! 798: $BB?9`<0(B
! 799: @item poly
! 800: $BB?9`<0(B
! 801: @end table
! 802:
! 803: @itemize @bullet
! 804: @item
! 805: $BM?$($i$l$?B?9`<0(B @var{poly} $B$KE,Ev$JM-M}?t$r3]$1$F(B, $B@0?t78?t$+$D(B
! 806: $B78?t$N(B GCD $B$,(B 1 $B$K$J$k$h$&$K$9$k(B.
! 807: @item
! 808: $BJ,?t$N;MB'1i;;$O(B, $B@0?t$N1i;;$KHf3S$7$FCY$$$?$a(B, $B<o!9$NB?9`<01i;;(B
! 809: $B$NA0$K(B, $BB?9`<0$r@0?t78?t$K$7$F$*$/$3$H$,K>$^$7$$(B.
! 810: @item
! 811: $BM-M}<0$rLsJ,$9$k(B @code{red()} $B$GJ,?t78?tM-M}<0$rLsJ,$7$F$b(B,
! 812: $BJ,;RB?9`<0$N78?t$OM-M}?t$N$^$^$G$"$j(B, $BM-M}<0$NJ,;R$r5a$a$k(B
! 813: @code{nm()} $B$G$O(B, $BJ,?t78?tB?9`<0$O(B, $BJ,?t78?t$N$^$^$N7A$G=PNO$5$l$k$?$a(B,
! 814: $BD>$A$K@0?t78?tB?9`<0$rF@$k;v$O=PMh$J$$(B.
! 815: @end itemize
! 816:
! 817: @example
! 818: [0] ptozp(2*x+5/3);
! 819: 6*x+5
! 820: [1] nm(2*x+5/3);
! 821: 2*x+5/3
! 822: @end example
! 823:
! 824: @table @t
! 825: @item $B;2>H(B
! 826: @fref{nm dn}.
! 827: @end table
! 828:
! 829: @node prim cont,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 830: @subsection @code{prim}, @code{cont}
! 831: @findex prim
! 832:
! 833: @table @t
! 834: @item prim(@var{poly}[,@var{v}])
! 835: :: @var{poly} $B$N86;OE*ItJ,(B (primitive part).
! 836: @item cont(@var{poly}[,@var{v}])
! 837: :: @var{poly} $B$NMFNL(B (content).
! 838: @end table
! 839:
! 840: @table @var
! 841: @item return poly
! 842: $BM-M}?t78?tB?9`<0(B
! 843: @item v
! 844: $BITDj85(B
! 845: @end table
! 846:
! 847: @itemize @bullet
! 848: @item
! 849: @var{poly} $B$N<gJQ?t(B ($B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v})
! 850: $B$K4X$9$k86;OE*ItJ,(B, $BMFNL$r5a$a$k(B.
! 851: @end itemize
! 852:
! 853: @example
! 854: [0] E=(y-z)*(x+y)*(x-z)*(2*x-y);
! 855: (2*y-2*z)*x^3+(y^2-3*z*y+2*z^2)*x^2+(-y^3+z^2*y)*x+z*y^3-z^2*y^2
! 856: [1] prim(E);
! 857: 2*x^3+(y-2*z)*x^2+(-y^2-z*y)*x+z*y^2
! 858: [2] cont(E);
! 859: y-z
! 860: [3] prim(E,z);
! 861: (y-z)*x-z*y+z^2
! 862: @end example
! 863:
! 864: @table @t
! 865: @item $B;2>H(B
! 866: @fref{var}, @fref{ord}.
! 867: @end table
! 868:
! 869: @node gcd gcdz,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 870: @subsection @code{gcd}, @code{gcdz}
! 871: @findex gcd
! 872:
! 873: @table @t
! 874: @item gcd(@var{poly1},@var{poly2}[,@var{mod}])
! 875: @item gcdz(@var{poly1},@var{poly2})
! 876: :: @var{poly1} $B$H(B @var{poly2} $B$N(B gcd.
! 877: @end table
! 878:
! 879: @table @var
! 880: @item return
! 881: $BB?9`<0(B
! 882: @item poly1,poly2
! 883: $BB?9`<0(B
! 884: @item mod
! 885: $BAG?t(B
! 886: @end table
! 887:
! 888: @itemize @bullet
! 889: @item
! 890: $BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B.
! 891: @item
! 892: @code{gcd()} $B$OM-M}?tBN>e$NB?9`<0$H$7$F$N(B GCD $B$rJV$9(B.
! 893: $B$9$J$o$A(B, $B7k2L$O@0?t78?t$G(B, $B$+$D78?t$N(B GCD
! 894: $B$,(B 1 $B$K$J$k$h$&$JB?9`<0(B, $B$^$?$O(B, $B8_$$$KAG$N>l9g$O(B 1 $B$rJV$9(B.
! 895: @item
! 896: @code{gcdz()} $B$O(B @var{poly1}, @var{poly2} $B$H$b$K@0?t78?t$N>l9g$K(B,
! 897: $B@0?t4D>e$NB?9`<0$H$7$F$N(B GCD $B$rJV$9(B.
! 898: $B$9$J$o$A(B, @code{gcd()} $B$NCM$K(B, $B78?tA4BN$N@0?t(B GCD$B$NCM$r3]$1$?$b$N$rJV$9(B.
! 899: @item
! 900: $B0z?t(B @var{mod} $B$,$"$k;~(B, @code{gcd()} $B$O(B GF(@var{mod}) $B>e$G$N(B GCD $B$rJV$9(B.
! 901: @item
! 902: @code{gcd()}, @code{gcdz()} Extended Zassenhaus $B%"%k%4%j%:%`$K$h$k(B.
! 903: $BM-8BBN>e$N(B GCD $B$O(B PRS $B%"%k%4%j%:%`$K$h$C$F$$$k$?$a(B, $BBg$-$JLdBj(B,
! 904: GCD $B$,(B 1 $B$N>l9g$J$I$K$*$$$F8zN($,0-$$(B.
! 905: @end itemize
! 906:
! 907: @example
! 908: [0] gcd(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
! 909: x^3+3*x^2+3*x+1
! 910: [1] gcdz(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
! 911: 6*x^3+18*x^2+18*x+6
! 912: [2] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y));
! 913: x^2-y^2
! 914: [3] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y),2);
! 915: x^3+y*x^2+y^2*x+y^3
! 916: @end example
! 917:
! 918: @table @t
! 919: @item $B;2>H(B
! 920: @fref{igcd igcdcntl}.
! 921: @end table
! 922:
! 923: @node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
! 924: @subsection @code{red}
! 925: @findex red
! 926:
! 927: @table @t
! 928: @item red(@var{rat})
! 929: :: @var{rat} $B$rLsJ,$7$?$b$N(B.
! 930: @end table
! 931:
! 932: @table @var
! 933: @item return
! 934: $BM-M}<0(B
! 935: @item rat
! 936: $BM-M}<0(B
! 937: @end table
! 938:
! 939: @itemize @bullet
! 940: @item
! 941: @b{Asir} $B$OM-M}?t$NLsJ,$r>o$K<+F0E*$K9T$&(B.
! 942: $B$7$+$7(B, $BM-M}<0$K$D$$$F$ODLJ,$O9T$&$,(B,
! 943: $BLsJ,$O%f!<%6!<$,;XDj$7$J$$8B$j9T$o$J$$(B.
! 944: $B$3$NLsJ,$r9T$&%3%^%s%I$,(B @t{red} $B$G$"$k(B.
! 945: @item
! 946: EZGCD $B$K$h$j(B @var{rat} $B$NJ,;R(B, $BJ,Jl$rLsJ,$9$k(B.
! 947: @item
! 948: $B=PNO$5$l$kM-M}<0$NJ,Jl$NB?9`<0$O(B, $B3F78?t$N(B GCD $B$,(B 1 $B$N(B
! 949: $B@0?t78?tB?9`<0$G$"$k(B.
! 950: $BJ,;R$K$D$$$F$O@0?t78?tB?9`<0$H$J$k$H$O8B$i$J$$(B.
! 951: @item
! 952: GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&DL0x;R$O2DG=$J8B$j=|$/$N$,(B
! 953: $BK>$^$7$$(B. $B$^$?(B, $BJ,Jl(B, $BJ,;R$,Bg$-$/$J$C$F$+$i$N$3$NH!?t$N8F$S=P$7$O(B,
! 954: $BHs>o$K;~4V$,3]$+$k>l9g$,B?$$(B. $BM-M}<01i;;$r9T$&>l9g$O(B, $B$"$kDxEY(B
! 955: $BIQHK$K(B, $BLsJ,$r9T$&I,MW$,$"$k(B.
! 956: @end itemize
! 957:
! 958: @example
! 959: [0] (x^3-1)/(x-1);
! 960: (x^3-1)/(x-1)
! 961: [1] red((x^3-1)/(x-1));
! 962: x^2+x+1
! 963: [2] red((x^3+y^3+z^3-3*x*y*z)/(x+y+z));
! 964: x^2+(-y-z)*x+y^2-z*y+z^2
! 965: [3] red((3*x*y)/(12*x^2+21*y^3*x));
! 966: (y)/(4*x+7*y^3)
! 967: [4] red((3/4*x^2+5/6*x)/(2*y*x+4/3*x));
! 968: (9/8*x+5/4)/(3*y+2)
! 969: @end example
! 970:
! 971: @table @t
! 972: @item $B;2>H(B
! 973: @fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}.
! 974: @end table
! 975:
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to poly.texi CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM / src / asir-doc / parts / builtin