File: [local] / OpenXM / src / k097 / lib / minimal / minimal-note-ja.txt (download)
Revision 1.8, Wed Jul 26 02:21:31 2000 UTC (23 years, 11 months ago) by takayama
Branch: MAIN
Changes since 1.7: +30 -4
lines
The construction algorithm of the schreyer frame in resol.c
has contained a bug. It has not constructed the minimal schreyer frame.
The bug is now fixed. See minimal-note-ja.txt for details and test data.
|
$OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.8 2000/07/26 02:21:31 takayama Exp $
SpairAndReduction() :
$BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.
V-minimal $B$KI,MW$+$I$&$+$NH=Dj$b$9$k(B.
SpairAndReduction2():
tower2 = StowerOf(tower,level-1);
SsetTower(tower2);
/** sm1(" show_ring "); */
$BM?$($i$l$?(B pair $B$r(B reduction $B$9$k$?$a$N(B schreyer order
$B$r@_Dj$9$k(B. Resolution $B$N?<$5$K1~$8$F(B, tower $B$b?<$/$9$kI,MW$,$"$k(B.
if (IsConstant(t_syz[i])){
Syzygy $B$r$_$F(B, $BDj?t@.J,$,$J$$$+(B check.
t_syz[i] $B$,Dj?t@.J,$G$"$l$P(B, $B0l$DA0$N(B GB $B$N9=@.MWAG$G$"$k(B
g_i $B$,M>J,$J(B GB $B$G$"$k2DG=@-$,$?$+$$(B.
SpairAndReduction() ( LaScala-Stillman $B$NJ}K!(B) $B$H$N@09g@-$r$H$k$?$a(B
g_i $B$r(B tmp[0] $B$KBeF~$7(B ( reduction $B$G$-$J$+$C$?$U$j$r$9$k(B )
g_i $B$N(B V-degree $B$r$7$i$Y$k(B.
Sannfs2_laScala2()
Sannfs3_laScala2() $B$r:n$k(B.
$BFs$D$N%"%k%4%j%:%`$NHf3S(B.
In(11)=sm1_pmat(a1[1]); $B$N=gHV$r$+$($k(B.
[
[ 3*Dx^2*h , 0 , Dy , -Dz ]
[ 6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy , 0]
[ 2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
[ 2*x*Dy*Dz , 0 , z , -y ]
[ 0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ]
]
In(12)=sm1_pmat(a2[1]);
[
[ 3*Dx^2*h , 0 , Dy , -Dz ]
[ 6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy, 0 ]
[ 2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
[ 2*x*Dy*Dz , 0 , z , -y ]
[ 9*z*Dx^2*h , 2*x*Dy^2*Dz-3*z*Dx^2*h , 3*z*Dy , 2*x*Dx ]
[ 2*x*Dx*Dz^2+3*z*Dz^3+5*Dz^2*h^2 , y*Dy*Dz^2-z*Dz^3-2*Dz^2*h^2 , 0 , 0 ]
]
In(13)=
----------------------
In(16)=sm1_pmat(a1[2]);
[
[ -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy^2 , 3*Dy*Dz , -2*x*Dy , 2*x*Dz , 0 ]
[ 3*y*z , z , y , -2*x*Dy*Dz , -3*z*Dy , 2*x*Dx , 2*x*z , -2*x*y , 0 ]
]
In(17)=sm1_pmat(a2[2]);
[
[ -y , 2*x*Dy*Dz , z , 0 , 2*x*Dx , 0 ]
[ -Dz , 3*Dx^2*h , Dy , -2*x*Dx-3*y*Dy-3*h^2 , -3*Dy*Dz , 0 ]
]
In(18)=
---------------------------
May 22, (Tue), 5:50 (Spain local time, 12:50 JST)
kan96xx/Kan/resol.c $B$G(B,
RemoveRedundantInSchreyerSkelton = 0
$B$KJQ$($F(B ($B$3$N(B option $B$b$"$?$i$7$/2C$($k(B), schreyer $B$,@5$7$/F0$/$+(B
$BD4$Y$k$3$H$K$9$k(B.
( commit $B$O(B kan96xx $B$H(B k097 $BN>J}$9$Y$7(B.)
test8() $B$G(B sm1 $B$G=q$$$?J}$N(B Schreyer $B$r8+$k$H(B,
RemoveRedundantInSchreyerSkelton = 1
$B$G$b(B,
kernel = image
$B$H$J$C$F$$$k$N$G0J8e$3$N(B option $B$O(B 1 $B$N$^$^;H$&$3$H$H$9$k(B.
$BMW$9$k$K(B k0 $B$N%3!<%I$,$I$&$d$i$*$+$7$$$i$7$$(B.
==>
6/8 $B$N%N!<%H$h$j(B.
syzygy $B$r(B homogenization $B$r2p$7$F7W;;$9$k$N$OLdBj$"$j(B.
--> usage of isExact
$BMW$9$k$K(B kernel = image $B$N%3!<%I$bJQ(B. Homogenized $B$N$^$^$d$kI,MW$"$j(B.
-----------------------------------
June 8, 2000 (Thu), 9:10 (Spain local time)
hol.sm1 : gb_h, syz_h, isSameIdeal, isSameIdeal_h
complex.sm1 : isExact, isExact_h
syzygy $B$r(B homogenization $B$r2p$7$F7W;;$9$k$N$OLdBj$"$j(B.
--> usage of isExact
[(Homogenize_vec) 0] system_variable : vector $B$N(B homogenize $B$r$7$J$$(B.
(grade) (module1v) switch_function : vector $BJQ?t$O(B, total
degree $B$K?t$($J$$(B.
==> $BL58B%k!<%W$KCm0U(B ---> gb_h, syz_h $B$N(B usage.
minimal-test.k $B$N(B ann(x^3-y^2*z^2) $B$N(B laplace $BJQ49$N(B
betti $B?t$,JQ(B, exact $B$G$J$$(B, $B$r(B isExact_h $B$G(B check
$B$7$h$&(B.
minimal-test.k
test10();
LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, schreyer resol $B$,(B exact $B$+(B
$BD4$Y$k(B.
$BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.
==> OK. IsExact_h $B$G$7$i$Y$k(B. (IsExact $B$O$@$a$h(B)
June 8, 2000 (Thu), 19:35
load["minimal-test.k"];;
test11();
LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, minimal resol $B$,(B exact $B$+(B
$BD4$Y$k(B.
$BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.
SwhereInTower $B$r;H$&$H$-$O(B,
SsetTower() $B$G(B gbList $B$rJQ99$7$J$$$H$$$1$J$$(B.
$B$b$A$m$s;HMQ$7$?$i(B, $B$=$l$rLa$9$3$H(B.
SpairAndReduction, SpairAndReduction2 $B$G(B,
SsetTower(StowerOf(tower,level));
pos = SwhereInTower(syzHead,tower[level]);
SsetTower(StowerOf(tower,level-1));
pos2 = SwhereInTower(tmp[0],tower[level-1]);
$B$H(B, SwhereInTower $B$NA0$K(B setTower $B$r$/$o$($?(B.
( $B0c$&%l%Y%k$G$NHf3S$N$?$a(B.)
IsExact_h $B$O(B, 0 $B%Y%/%H%k$r4^$`>l9g(B, $B$?$@$7$/F0:n$7$J$$$h$&$@(B.
test11().
test11a() $B$G(B, 0 $B%Y%/%H%k$r<j$G=|$$$?9TNs$N(B exactness $B$r%A%'%C%/(B. ==> OK.
---------------------------------
June 9, 6:20
SpairAndReduction
$B$H(B
SpairAndReduction2
$B$N0c$$(B.
SpairAndReduction : SlaScala (LaScala-Stillman's algorithm $B$G;H$&(B)
SpairAndReduction2 : Sschreyer (schreyer algorithm $B$G;H$&(B, laScala $B$O$J$7(B.)
0 $B$r<+F0$G=|$/%3!<%I$r=q$3$&(B.
SpruneZeroRow() $B$r(B Sminimal() $B$K2C$($?(B.
test11() $B$b@5$7$/F0:n$9$k$O$:(B.
IsExact_h $B$O(B schreyer $B$r(B off $B$7$F(B, ReParse $B$7$F$+$i(B,
$B8F$S=P$9$3$H(B.
#ifdef TOTAL_STRATEGY
return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
#endif
/* Strategy must be compatible with ordering. */
/* Weight vector must be non-negative, too. */
/* See Sdegree, SgenerateTable, reductionTable. */
wd = Sord_w(f,ww);
return(wd+Sdegree(tower[level-2,i],tower,level-1));
TOTAL_STRATEGY $B$rMQ$$$kI,MW$,$"$k$N$G$O(B??
Example 1: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
v=[[2*x*Dx + 3*y*Dy+6, 0],
[3*x^2*Dy + 2*y*Dx, 0],
[0, x^2+y^2],
[0, x*y]];
a=Sminimal(v);
strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B?
a=test_ann3("x^3+y^3+z^3); $B$O;~4V$,$+$+$j$=$&(B.
a=test_ann3("x^3+y^3"); OK.
a=test_ann3("x^2+y^2+z"); OK.
$B>e$N(B example 1 $B$N%(%i!<(B $B$N8+J}(B:
Processing [ 1 , 3 ] Strategy = 2
1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B.
In(7)=reductionTable:
[[ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ]
-- $B$3$l(B.
SpairAndReduction:
[ p and bases , [ [ 0 , 3 ] , [ y*h , -x ] ] , [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ] ]
0 $B$N(B 0 $BHVL\$H(B 3 $BHVL\(B $B$N(B spair $B$r7W;;$7$F(B, 0 $B%l%Y%k$N(B gb $B$G(B reduction.
[ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B.
( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. )
[ level= , 1 ]
[ tower2= , [ [ ] ] ] ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. )
[ y*h , -es^3*x ]
[gi, gj] = [ 2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ]
1
Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy
by [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]
result is [ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [ 0 , 0 , 0 , 0 , 0 , 0 ] ]
vdegree of the original = -1
vdegree of the remainder = -1
[ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [ y*h , 0 , 0 , -x , 0 , 0 ] , 3 , 5 , -1 , -1 ]
In(11)=freeRes:
[ [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ] , [ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ] , [ %[null] ] ]
$B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B
$B$i$l$?(B.
$B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B.
i = SnextI(reductionTable_tmp,strategy,redundantTable,
skel,level,freeRes);
In(22)=reductionTable:
[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ]
$B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B,
In(25)=skel[2]:
[ [ [ 0 , 2 ] , [ 1 , -y^2 ] ] ]
$B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair.
$B$7$+$7(B,
In(26)=bases:
[ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ]
$B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B.
reductionTable_tmp=[ 2 ]
See also reductionTable, strategy, level,i
ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
--- Engine error or interrupt : In function : Error of class PrimitiveObject
Type in Cleards() to exit the debug mode and Where() to see the stack trace.
In(7)=reductionTable:
[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ]
In(8)=strategy:
2
In(9)=level:
2
RemoveRedundantInSchreyerSkelton = 0
$B$H$7$F$bF1$8%(%i!<(B.
-------------------------------------------------
test_ann3("x*y+y*z+z*x"); OK.
6/9 (Fri)
Sminimal $B$N<BAu$KAjJQ$o$i$:6lO+$7$F$^$9(B.
Sevilla $B$G$$$m$$$m$HD>$7$?7k2L(B,
Sminimal $B$O$&$^$/$&$4$1$P@5$7$$Ez$($r$@$7$F$k$_$?$$$G$9$,(B
(D<h> : homogenized Weyl $B$G(B ker = im $B$r(B check $B$7$F$k(B,
V-adapted (strict) $B$+$I$&$+$N(B check routing $B$O$^$@=q$$$F$J$$(B),
strategy $B$,$&$^$/$&$4$+$J$/$F$H$^$k>l9g$b$"$j$^$9(B
( strategy = 2 $B$N(B sp $B$r7W;;$9$k$N$K(B, strategy 3 $B$N(B $B85$rI,MW$H(B
$B$7$?$j$9$k>l9g$"$j(B).
strategy $B$O(B
def Sdegree(f,tower,level) {
local i,ww, wd;
/* extern WeightOfSweyl; */
ww = WeightOfSweyl;
f = Init(f);
if (level <= 1) return(StotalDegree(f));
i = Degree(f,es);
return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
}
$B$rMQ$$$F(B,
ans_at_each_floor[j] = Sdegree(tower[i,j],tower,i+1)-(i+1)
$B$G7W;;$7$F$^$9(B.
$B$$$/$D$+=PNO$r$D$1$F$*$-$^$9$N$G(B, $B8!F$(B!!!
$BNc(B 1:
load["minimal-test.k"];;
a=test_ann3("x^3-y^2*z^2"); $B0z?t$N(B annihilating ideal $B$N(B laplace $BJQ49$N(B
homogenization $B$N(B resolution.
weight vector $B$O(B (-1,-1,-1,1,1,1)
In(4)=sm1_pmat(a[1]);
[
[ 0 $B<!(B
[ y*Dy-z*Dz ]
[ -2*x*Dx-3*z*Dz+h^2 ]
[ 2*x*Dy*Dz^2-3*y*Dx^2*h ]
[ 2*x*Dy^2*Dz-3*z*Dx^2*h ]
]
[ 1 $B<!(B
[ 3*Dx^2*h , 0 , Dy , -Dz ]
[ 6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy , 0 ]
[ 0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ]
[ 2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
[ 2*x*Dy*Dz , 0 , z , -y ]
]
[ 2 $B<!(B
[ -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy*Dz ]
[ 3*y*z , z , y , -2*x*Dy*Dz , 2*x*Dx ]
]
]
In(5)=
$BNc(B 2:
load["minimal-test.k"];;
a=test_ann3("x*y+y*z+z*x");
In(6)=sm1_pmat(a[1]);
[
[ 0 $B<!(B
[ 2*x*Dx+x*Dz-y*Dz+z*Dz+h^2 ]
[ -2*y*Dy+x*Dz-y*Dz-z*Dz-h^2 ]
[ -2*x*Dy+2*z*Dy+x*Dz-y*Dz+3*z*Dz+h^2 ]
[ -2*y*Dx+2*z*Dx-x*Dz+y*Dz+3*z*Dz+h^2 ]
]
[ 1 $B<!(B
[ y-z , x-z , -y , x ]
[ 2*Dy-2*Dz , 2*Dx-2*Dz , 2*Dx+2*Dz , -2*Dy-2*Dz ]
[ 2*y*Dx-2*z*Dx+x*Dz-y*Dz-3*z*Dz-2*h^2 , 0 , 0 , 2*x*Dx+x*Dz-y*Dz+z*Dz+2*h^2 ]
[ 2*y*Dy-2*z*Dy+y*Dz-z*Dz+h^2 , 2*x*Dz-y*Dz+2*z*Dz+h^2 , -x*Dz+z*Dz , 2*x*Dy+x*Dz ]
[ -2*y*Dy+2*z*Dy+y*Dz-z*Dz , y*Dz-4*z*Dz , -2*y*Dx+2*z*Dx-y*Dz+2*z*Dz , -2*z*Dy+y*Dz-3*z*Dz ]
]
[ 2 $B<!(B
[ -2*y*Dx+2*z*Dx-y*Dz+2*z*Dz , x*y-x*z-y*z+z^2 , y-z , y , x+y-z ]
[ -6*Dx*Dz-2*Dz^2 , x*Dz+y*Dz-5*z*Dz-4*h^2 , -2*Dy+2*Dz , 2*Dx+2*Dz , 4*Dz ]
]
]
In(7)=
$BNc(B 3: $B$&$^$/9T$+$J$$Nc(B:
Example 1: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
v=[[2*x*Dx + 3*y*Dy+6, 0],
[3*x^2*Dy + 2*y*Dx, 0],
[0, x^2+y^2],
[0, x*y]];
a=Sminimal(v);
strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B?
Negative weight vector $B$r;H$o$J$$$H$-$A$s$HF0$-$^$9(B.
DEBUG $B=PNO(B:
rf= [
[
[ Schreyer frame.
[ 0 , y^3 , 0 , 0 , -x^2 , 0 ]
[ 0 , 0 , y^2 , 0 , -x , 0 ]
[ 0 , y , -x , 0 , 0 , 0 ]
[ y*h , 0 , 0 , -x , 0 , 0 ]
[ 0 , 0 , 0 , 3*y*Dy , 0 , -2*Dx ]
]
[
[ 1 , 0 , -y^2 , 0 , 0 ]
]
[ ]
]
[
[ 2*x*Dx , e_*x^2 , e_*x*y , 2*y*Dx*h , e_*y^3 , 3*y^2*Dy*h ]
[ es*y^3 , es^2*y^2 , es*y , y*h , 3*es^3*y*Dy ]
[ 1 ]
]
[
[ ]
[
[
[ 1 , 4 ]
[ y^3 , -x^2 ]
]
[
[ 2 , 4 ]
[ y^2 , -x ]
]
[
[ 1 , 2 ]
[ y , -x ]
]
[
[ 0 , 3 ]
[ y*h , -x ]
]
[
[ 3 , 5 ]
[ 3*y*Dy , -2*Dx ]
]
]
[
[
[ 0 , 2 ]
[ 1 , -y^2 ]
]
]
[ ]
]
[ resolution $B$9$Y$-(B $BItJ,2C72(B e_ $B$O(B $B%Y%/%H%k@.J,$N%^!<%/(B.
[ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]
]
]
$BN,(B
Processing [ 1 , 3 ] Strategy = 2
1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B.
In(7)=reductionTable:
[[ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ]
-- $B$3$l(B.
SpairAndReduction:
[ p and bases , [ [ 0 , 3 ] , [ y*h , -x ] ] , [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ] ]
0 $B$N(B 0 $BHVL\$H(B 3 $BHVL\(B $B$N(B spair $B$r7W;;$7$F(B, 0 $B%l%Y%k$N(B gb $B$G(B reduction.
[ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B.
( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. )
[ level= , 1 ]
[ tower2= , [ [ ] ] ] ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. )
[ y*h , -es^3*x ]
[gi, gj] = [ 2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ]
1
Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy
by [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]
result is [ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [ 0 , 0 , 0 , 0 , 0 , 0 ] ]
vdegree of the original = -1
vdegree of the remainder = -1
[ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [ y*h , 0 , 0 , -x , 0 , 0 ] , 3 , 5 , -1 , -1 ]
In(11)=freeRes:
[ [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ] , [ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ] , [ %[null] ] ]
$B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B
$B$i$l$?(B.
$B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B.
i = SnextI(reductionTable_tmp,strategy,redundantTable,
skel,level,freeRes);
In(22)=reductionTable:
[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ]
$B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B,
In(25)=skel[2]:
[ [ [ 0 , 2 ] , [ 1 , -y^2 ] ] ]
$B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair.
$B$7$+$7(B,
In(26)=bases:
[ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ]
$B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B.
reductionTable_tmp=[ 2 ]
See also reductionTable, strategy, level,i
ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
--- Engine error or interrupt : In function : Error of class PrimitiveObject
Type in Cleards() to exit the debug mode and Where() to see the stack trace.
In(7)=reductionTable:
[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ]
In(8)=strategy:
2
In(9)=level:
2
$B$3$N;~E@$^$G$G$b$H$^$C$?(B basis
[
[ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]
[ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ]
[ %[null] ]
]
-------------------------------------
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
a=Sminimal([x^2+y^2,x*y]);
$B$3$l$G$b;w$?$h$&$J%(%i!<$r$@$;$k(B.
$B$3$NJ}$,(B debug $B$7$d$9$$(B:
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
a=Sminimal([x*y,x^2+y^2]);
$B$G$O%(%i!<$,$G$J$$$N$,IT;W5D(B.
pruneZero $B$,F0$$$F$J$$$N$,JQ(B.
rf= [
[
[
[ y^3 , 0 , -x^2 ]
[ 0 , y^2 , -x ]
[ y , -x , 0 ]
]
[
[ 1 , 0 , -y^2 ]
]
[ ]
]
[
[ x^2 , x*y , y^3 ]
[ y^3 , es*y^2 , y ]
[ 1 ]
]
[
[ ]
[
[
[ 0 , 2 ]
[ y^3 , -x^2 ]
]
[
[ 1 , 2 ]
[ y^2 , -x ]
]
[
[ 0 , 1 ]
[ y , -x ]
]
]
[
[
[ 0 , 2 ]
[ 1 , -y^2 ]
]
]
[ ]
]
[
[ x^2+y^2 , x*y , y^3 ]
]
]
[ 0 , 0 ]
Processing [ 0 , 0 ] Strategy = 1
[ 0 , 1 ]
Processing [ 0 , 1 ] Strategy = 1
[ 1 , 2 ]
Processing [ 1 , 2 ] Strategy = 1
SpairAndReduction:
[ p and bases , [ [ 0 , 1 ] , [ y , -x ] ] , [ x^2+y^2 , x*y , %[null] ] ]
[ level= , 1 ]
[ tower2= , [ [ ] ] ]
[ y , -es*x ]
[gi, gj] = [ x^2+y^2 , x*y ]
1
Reduce the element y^3
by [ x^2+y^2 , x*y , %[null] ]
result is [ y^3 , 1 , [ 0 , 0 , 0 ] ]
vdegree of the original = -3
vdegree of the remainder = -3
[ y^3 , [ y , -x , 0 ] , 2 , 2 , -3 , -3 ]
[ 0 , 2 ]
Processing [ 0 , 2 ] Strategy = 2
[ 1 , 1 ]
Processing [ 1 , 1 ] Strategy = 2
SpairAndReduction:
[ p and bases , [ [ 1 , 2 ] , [ y^2 , -x ] ] , [ x^2+y^2 , x*y , y^3 ] ]
[ level= , 1 ]
[ tower2= , [ [ ] ] ]
[ es*y^2 , -es^2*x ]
[gi, gj] = [ x*y , y^3 ]
1
Reduce the element 0
by [ x^2+y^2 , x*y , y^3 ]
result is [ 0 , 1 , [ 0 , 0 , 0 ] ]
vdegree of the original = -4
vdegree of the remainder = %[null]
[ 0 , [ 0 , y^2 , -x ] , 1 , -1 , -4 , %[null] ]
reductionTable_tmp=[ 2 ]
See also reductionTable, strategy, level,i
ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
--- Engine error or interrupt : In function : Error of class PrimitiveObject
Type in Cleards() to exit the debug mode and Where() to see the stack trace.
In(10)=reductionTable :
[ [ 1 , 1 , 2 ] , [ 3 , 2 , 1 ] , [ 2 ] ]
In(11)=bases:
[ %[null] , [ 0 , y^2 , -x ] , [ -y , x , 1 ] ]
In(12)= $B$3$l$O(B, [3, 2, 1] $B$N85$N$&$A(B, 2,1 $B$,$b$H$^$C$F$$$k(B.
$B:G8e$N(B [ 2 ] $B$N7W;;$K(B 0 $BHVL\$,I,MW$G$3$l$,$^$@$J$$(B.
$BMW$9$k$K(B 1 $BHVL\$H(B 3 $BHVL\$r>C$9(B operator [1, 0, -y^2]
[ y^3 , 0 , -x^2 ]
[ 0 , y^2 , -x ]
[ y , -x , 0 ]
$B$N(B reduction $B$,I,MW(B.
-----------------------------------------
June 11, 2000 (Tue), 20:05
V-strict $B$+$I$&$+$r%A%'%C%/$9$k4X?t$r=q$-$?$$(B.
$B0BA4$K(B ring (schreyer order) $B$rDj5A$9$k4X?t$,M_$7$$(B.
$B0BA4$K(B parse $B$9$k4X?t$bM_$7$$(B.
$B%Y%/%H%k$H(B es $BI=8=$NJQ494X?t$b$$$k(B.
AvoidTheSameRing == 1 $B$J$i(B, schreyer $B$N(B gbList $B$bJQ99$G$-$J$$$h$&$K(B
$B$9$Y$-$+!)(B
$B4XO"JQ?t(B:
needWarningForAvoidTheSameRing
isTheSameRing() : ring $B$,F1$8$+(B check. pointer $B$G$J$/Cf?H$^$G$_$k(B.
see poly4.c. $B$3$3$N%3%a%s%H$O;29M$K$J$k(B.
3.If Schreyer = 1, then the system always generates a new ring.
define_ring $B$K(B gbList $B$bEO$;$k$N(B?
==> set_up_ring@ $B$r8+$k(B. grep set_up_ring ==>
primitive.c KsetUpRing() grep KsetUpRing ==>
keyword gbListTower $B$,;H$($k$,(B, list $B$GM?$($J$$$H$$$1$J$$(B.
list $B$KJQ49$9$k$N$O(B, (list) dc.
tparse $B$NI,MW$J$o$1(B?
?? $B$*$b$$$@$;$J$$(B.
ring_def $B$G(B ring (schreyer order) $B$rDj5A$9$k$H(B, $B7W;;$N$H$-$N(B
order $B$b(B tower $B$G$d$C$F$/$l$k$N(B?
$BB?J,(B NO.
grep ppAdd *.c ==>
poly2.c
checkRing(f,g);
while (f != POLYNULL && g != POLYNULL) {
/*printf("%s + %s\n",POLYToString(f,'*',1),POLYToString(g,'*',1));*/
checkRing2(f,g); /* for debug */
gt = (*mmLarger)(f,g);
mmLarger $B$OJQ$($F$J$$$h$&$K8+$($k(B. checkRing $B$O%^%/%m(B.
mmLarger_tower $B$O(B
if (!(f->m->ringp->schreyer) || !(g->m->ringp->schreyer))
return(mmLarger_matrix(f,g));
$B$H$J$C$F$k$N$G(B mmLarger_tower $B$r(B default $B$K$7$F$*$1$P?4G[$J$$$h$&$K8+$($k(B.
ring_def $B$O@5$7$/F0$/(B?
TODO:
$B4X?t$N;EMM(B: ( new.sm1 $B$^$?$O(B complex.sm1 $B$K$*$$$H$/(B )
mmLarger $B$O(B tower $B$KJQ$($F$7$^$&(B.
$BJQ?tL>(B, weight vector, $B%7%U%H%Y%/%H%k(B m $B$rM?$($k$H(B ring (with schreyer order)
$B$r:n$k(B. ==> weyl<m>, weyl
parser $B$O$H$/$K:n$kI,MW$,$J$$$h$&$K8+$($k$,(B...(tparse) ==> name
$B%Y%/%H%k(B <---> es $BI=8=(B cf. toVectors, [(toe_) f] gbext ==> name
$BE,@Z$J(B homogenization $B4X?t(B ==> homogenize<m>
ord_w $B$N(B schreyer $BHG(B ==> ord_w<m>
init $B$N(B schreyer $BHG(B ==> init<m>
gb_h, syz_h $B$NBP1~HG(B ==> [ ii vv ww m] syz_h
resolution $B$+$i(B shift vector $B$r7W;;$9$k4X?t(B.
$B7k2L$N(B check $B$r$9$k(B assert $B4X?t$bI,MW(B.
$B>e$N(B $B%7%U%H%Y%/%H%kBP1~HG$N4X?t$OEvJ,(B new.sm1 $B$X(B. $B$=$N$"$H(B complex.sm1 $B$X(B.
cohom.sm1 $B$N(B interface $B4X?t$O(B cohom.k $B$X(B.
Help key word $B$O(B (Cohom.deRham) $B$_$?$$$K(B, . $B$G$/$.$C$F=q$/(B.
----------------------
$B%(%i!<$N860x$,$h$&$d$/$o$+$k(B: June 14, 19:00
Schreyer frame $B$NCJ3,$G(B syz $B$K(B 1 $B$,$"$k$H(B strategy $B$,(B
$B$O$?$i$+$J$$(B.
test13() GKZ $B$N(B minimal free resolution. 2 $BEY<B9T$9$k$HJQ(B.
grade $B$,JQ99$5$l$k$H(B, $BJQ$J$3$H$,$*$-$k$N$G(B,
ScheckIfSchreyer() $B4X?t$G(B, $B$3$l$r(B scheck $B$9$k$3$H$K$7$?(B.
sm1(" (report) (mmLarger) switch_function /ss set ");
$B$O$^$@$d$a$H$/(B. matrix $B$K$J$C$F$k$N$G(B.
------------------------------------------
June 15, 2000
TODO:
1.if (IdenfityIntegerAndUniversalNumber) $B$N$H$-(B --- default
lt, gt, eq $B$G(B integer $B$H(B universalNumber $B$NHf3S$,$G$-$k$h$&$K$9$k(B.
rational $B$H$NHf3S$b2DG=$K$9$k(B.
2. sm1_push_int0 $B$KBP1~$9$k$3$H$r(B, sm1 $B$NB&$G$d$k(B.
$B%^%/%mL>(B obj to_int --> Done.
weight_vector $B$N(B universalNumber ==> $B$^$@(B. $B%(%i!<$r$@$5$J$$$N$,$3$o$$(B.
s_weight_vector
weightv
ord_w
toVectors
define_ring
init
gkz
-------------
Schreyer skelton $B$,$I$&$7$F(B 1 $B$rMWAG$K$b$D$+$7$i$Y$k(B.
June 24 (Sat), 22:30 at Posthouse (Heathrow) www.posthouse-hotels.com
Sevilla $BBZ:_(B, Mega $B$b$h$&$d$/$*$o$j(B minimal resolution $B$N(B check $B$KLa$k(B.
resol1.c $B$K<!$N(B line $B$r2C$($?(B.
/* If isConstant(sv.a) is added, (x^3 - y^2 z^2) deRham stops
with an error. I've not yet understood the reason.
At Posthouse at Heathrow. June 24, 2000 */
if (isConstant(sv.b)) {
s->deleted = 1;
}
===> $B$*$+$7$$$N$G:o=|(B.
isConstant(sv.a) $B$,$J$$$H(B, $B$3$s$I$O(B,
Sminimal([x^2+y^2,x*y]); $B$,%(%i!<$G$H$^$k(B.
(x,y $B$N(B weight $B$O(B -1).
LaScala-Stillman $B$NO@J8$r$b$&0lEY$J$,$a$h$&(B.
commit $B$9$Y$-(B: misc/mega2000 (cvs-misc add) Done.
OpenXM/src/kan96xx Done.
OpenXM/src/k097/lib/minimal Done.
July 26.
resol.c $B$N(B schreyerSkelton0 $B$G(B, skelton $B$,(B minimal $B$K$J$k$h$&$K(B
$B%3!<%I$rA^F~(B.
$B%F%9%H$O(B
cd src/k097/lib/minimal
k0
load["minimal.k"];;
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
Sminimal([x^2+y^2,x*y]);
$B$G(B.
LaScala-Stillman $B$NO@J85U$G(B i<j $B$J$i(B e_i > e_j $B$H$J$k(B.
(order.c mmLarger_tower())
$B%F%9%H(B 2.
cd src/k097/lib/minimal
k0
load["minimal-test.k"];;
v:
Sminimal(v);
test11(); /* a = test_ann3("x^3-y^2*z^2"); */
test14(); /* gkz (1,2,3) */
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