File: [local] / OpenXM / src / k097 / lib / minimal / minimal-note-ja.txt (download)
Revision 1.12, Wed Aug 9 03:45:27 2000 UTC (23 years, 10 months ago) by takayama
Branch: MAIN
CVS Tags: maekawa-ipv6, R_1_3_1-2, RELEASE_1_3_1_13b, RELEASE_1_2_3_12, RELEASE_1_2_3, RELEASE_1_2_2_KNOPPIX_b, RELEASE_1_2_2_KNOPPIX, RELEASE_1_2_2, RELEASE_1_2_1, RELEASE_1_1_3, KNOPPIX_2006, HEAD, DEB_REL_1_2_3-9
Changes since 1.11: +433 -1
lines
Update of the documentation for minimal.k (written at At.Andrews).
|
$OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.12 2000/08/09 03:45:27 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$$$H(B, $B$3$s$I$O(B,
Sminimal([x^2+y^2,x*y]); $B$,%(%i!<$G$H$^$k(B.
(x,y $B$N(B weight $B$O(B -1).
LaScala-Stillman $B$NO@J8$r$b$&0lEY$J$,$a$h$&(B.
commit $B$9$Y$-(B: misc/mega2000 (cvs-misc add) Done.
OpenXM/src/kan96xx Done.
OpenXM/src/k097/lib/minimal Done.
July 26.
resol.c $B$N(B schreyerSkelton0 $B$G(B, skelton $B$,(B minimal $B$K$J$k$h$&$K(B
$B%3!<%I$rA^F~(B.
$B%F%9%H$O(B
cd src/k097/lib/minimal
k0
load["minimal.k"];;
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
Sminimal([x^2+y^2,x*y]);
$B$G(B.
LaScala-Stillman $B$NO@J85U$G(B i<j $B$J$i(B e_i > e_j $B$H$J$k(B.
(order.c mmLarger_tower())
$B%F%9%H(B 2.
cd src/k097/lib/minimal
k0
load["minimal-test.k"];;
v:
Sminimal(v);
test11(); /* a = test_ann3("x^3-y^2*z^2"); */
test14(); /* gkz (1,2,3) */
July 30. Removed unnecessary code.
$BNc(B:
Sminimal("x^3-y^2");
test12() ( x^3-y^2 z^2)
test15() GKZ 1,2,3 with a check.
test15b() toric
test15c() (u,v) = (-1,1)
August 1.
(u,v)-minimal $B$N%F%9%H%3!<%I$r$$$l$?(B.
IsExact_h $B$G(B $BJQ?t(B c $B$NCM$,$+$o$k(B. $B860xITL@(B.
c=Sinit_w(b,w);
Println("Resolution (b)----");
sm1_pmat(b);
Println("Initial (c)----");
sm1_pmat(c); cc=c;
Println("Exactness of the resolution ---");
Println(IsExact_h(b,v)); /* IsExact_h breaks the variable c.
THIS BUG SHOULD BE FIXED. */
$B$3$N$"$H$J$<$+(B, c $B$,(B b $B$NCM$K$+$o$C$F$7$^$&(B.
$B$J$*(B def IsExact(c,...) $B$HDj5A$5$l$F$*$j(B, $B$3$N(B c $B$rJL$NJQ?tL>$K(B
$BJQ$($l$P$3$NLdBj$O$*$-$J$$(B.
Println("Why is the initial c rewritten by b? (buggy) ");sm1_pmat(c[0]);
===> complex.sm1 $B$N(B isExact_h (isExact) $B$G(B popVariables $B$rK:$l$F$?$@$1(B.
betti $B?t$O(B, $B9TNs$N>C5n$r$d$k$^$G$o$+$i$J$$$N(B?
SbettiTable().
Sminimal $B$O(B [(Homogenize_vec) 0] system_variable $B$K$9$k$h$&$G(B,
$B$3$l$,(B cohomology $B$N7W;;$K$O<YKb(B.
August 2, 2000.
Sminimal $B$O(B [(Homogenize_vec) 0] system_variable $B$K$9$k$h$&$G(B,
$B$3$l$,(B cohomology $B$N7W;;$K$O<YKb(B.
( cf. $BBg0$5W;a$N%9%/%j%W%H(B. $B8=:_?@8M$KBZ:_Cf(B. )
/restoreEnvAfterResolution {
[(AvoidTheSameRing)] pushEnv
[ [(AvoidTheSameRing) 0] system_variable
[(gbListTower) [[ ]] (list) dc] system_variable
] pop popEnv
setupEnvForResolution.opts restoreOptions <=== $BJQ99(B. opts $B$O$$$m$s$J$H$3$m$G;H$C$F$k(B.
} def
$B$3$N%^%/%m$r$h$Y$P$$$$$N$+!)(B
sm1(" restoreEnvAfterResolution ");
$B$r(B Sminimal $B$N$*$o$j$K8F$V$h$&$KJQ$($?(B.
test17b(), test18() $B$O@5>oF0:n(B.
August 7, Mon 13:00JST ( 5:00 St.Andrews, Scotland, 4039 $B9f<<(B)
example-ja.tex $B$r=q$/$?$a$N=PNO(B.
% k0
sm1>macro package : dr.sm1, 9/26,1995 --- Version 6/15, 2000.
sm1>macro package : module1.sm1, 1994 -- Nov 8, 1998
This is kan/k0 Version 1998,12/15
WARNING: This is an EXPERIMENTAL version
sm1>var.sm1 : Version 3/7, 1997
In(1)=Loading startup files (startup.k) 1997, 3/11.
sm1 version = 3.000726
Default ring is Z[x,h].
WARNING(sm): You rewrited the protected symbol pushVariables.
WARNING(sm): You rewrited the protected symbol popVariables.
In(2)=load["minimal-test.k"];;
cpp: -lang-c++: linker input file unused since linking not done
cpp: -lang-c++: linker input file unused since linking not done
cohom.sm1 is the top of an experimental package to compute restrictions
of all degrees based on restall.sm1 and restall_s.sm1
See, http://www.math.kobe-u.ac.jp to get these files of the latest version.
Note that the package b-function.sm1 cannot be used with this package.
r-interface.sm1 (C) N.Takayama, restriction, deRham
oxasir.sm1, --- open asir protocol module 3/1 1998, 6/5 1999
asirconnect, asir, fctr, primadec, (C) M.Noro, N.Takayama
ox.sm1, --- open sm1 protocol module 11/11,1999 (C) N.Takayama. oxhelp for help
hol.sm1, basic package for holonomic systems (C) N.Takayama, 2000, 06/08
rank characteristic ch rrank gb pgb syz genericAnn annfs gb_h syz_h isSameIdeal isSameIdeal_h
sm1>gkz.sm1 generates gkz systems (C) N.Takayama, 1998, 11/8, cf. rrank in hol.sm1
gkz
sm1>appell.sm1 generates Appell hypergeometric differential equations (C) N.Takayama, 1998, 11/8, cf. rank in hol.sm1
appell1 appell4
sm1>resol0.sm1, package to construct schreyer resolutions -- not minimal
(C) N.Takayama, 1999, 5/18. resol0, resol1
complex.sm1 : 1999, 9/28, res-div, res-solv, res-kernel-image, res-dual
2000, 6/8, isExact_h, isExact
In this package, complex is expressed in terms of matrices.
restall.sm1 ... compute all the cohomology groups of the restriction
of a D-module to tt = (t_1,...,t_d) = (0,...,0).
non-Schreyer Version: 19980415 by T.Oaku
usage: [(P1)...] [(t1)...] bfm --> the b-function
[(P1)...] [(t1)...] k0 k1 deg restall --> cohomologies of restriction
[(P1)...] [(t1)...] intbfm --> the b-function for integration
[(P1)...] [(t1)...] k0 k1 deg intall --> cohomologies of integration
restall_s.sm1...compute all the cohomology groups of the restriction
of a D-module to tt = (t_1,...,t_d) = (0,...,0).
Schreyer Version: 19990521 by N.Takayama & T.Oaku
usage: [(P1)...] [(t1)...] k0 k1 deg restall_s -> cohomologies of restriction
[(P1)...] [(t1)...] k0 k1 deg intall_s --> cohomologies of integration
No truncation from below in restall
The variable Schreyer is set to 2.
Loading tower.sm1 in the standard context. You cannot use Schyrer 1. It is controlled from cohom.sm1
oxpath.oxlog.xterm is set to /home/nobuki/OpenXM/lib/sm1/bin/oxlog
In(3)=a=Sannfs2("x^3-y^2");
Starting ox_asir server.
Hello from open. serverName is localhost and portnumber is 0
Done the initialization. port =1024
Hello from open. serverName is localhost and portnumber is 0
Done the initialization. port =1025
[ 7 , 1025 , 6 , 1024 ]
[1] 250
Trying to accept from localhost... len= 16
4 2 7f 0 0 1 0 0 0 0 0 0 0 0 8 0
Authentification: localhost is allowed to be accepted.
Accepted.
Trying to accept from localhost... len= 16
4 3 7f 0 0 1 0 0 0 0 0 0 0 0 6 0
Authentification: localhost is allowed to be accepted.
Accepted.
Control port 1024 : Connected.
Stream port 1025 : Connected.
Byte order for control process is network byte order.
Byte order for engine process is network byte order.
WeightOfSweyl=[ x , -1 , y , -1 , Dx , 1 , Dy , 1 ]
Automatic homogenization.
[ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
....Done. betti=4
Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
.Done. betti=1
Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
Warning: Homogenization and ReduceLowerTerms options are automatically turned off.
Done. betti=0
Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.
rf= [
[
[
[ -9*y^2*Dy , 0 , 2*x , 0 ]
[ 0 , 0 , -3*y*Dy , Dx ]
[ 0 , -3*y*Dy , Dx , 0 ]
[ -3*y*Dx , 2*x , 0 , 0 ]
]
[
[ -Dx , 0 , 0 , 3*y*Dy ]
]
[ ]
]
[
[ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
[ -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
[ -Dx ]
]
[
[ ]
[
[
[ 0 , 2 ]
[ -9*y^2*Dy , 2*x ]
]
[
[ 2 , 3 ]
[ -3*y*Dy , Dx ]
]
[
[ 1 , 2 ]
[ -3*y*Dy , Dx ]
]
[
[ 0 , 1 ]
[ -3*y*Dx , 2*x ]
]
]
[
[
[ 0 , 3 ]
[ -Dx , 3*y*Dy ]
]
]
[ ]
]
[
[ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , -27*y^3*Dy^2-27*y^2*Dy*h^2+3*y*h^4+8*x^3*Dy*h ]
]
]
Generating reduction table which gives an order of reduction.
WeghtOfSweyl=[ x , -1 , y , -1 , Dx , 1 , Dy , 1 ]
tower[ [ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ] , [ -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ] , [ -Dx ] ]
reductionTable= [
[ 1 , 2 , 3 , 4 ]
[ 3 , 4 , 3 , 2 ]
[ 3 ]
]
[ 0 , 0 ]
Processing [level,i]= [ 0 , 0 ] Strategy = 1
[ 0 , 1 ]
Processing [level,i]= [ 0 , 1 ] Strategy = 2
[ 1 , 3 ]
Processing [level,i]= [ 1 , 3 ] Strategy = 2
SpairAndReduction:
[ p and bases , [ [ 0 , 1 ] , [ -3*y*Dx , 2*x ] ] , [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ] ]
[ level= , 1 ]
[ tower2= , [ [ ] ] ]
[ -3*y*Dx , 2*es*x ]
[gi, gj] = [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h ]
1
Reduce the element 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h
by [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]
result is [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 1 , [ 0 , 0 , 0 , 0 ] ]
vdegree of the original = 0
vdegree of the remainder = 0
[ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , [ -3*y*Dx , 2*x , 0 , 0 ] , 3 , 2 , 0 , 0 ]
[ 0 , 2 ]
Processing [level,i]= [ 0 , 2 ] Strategy = 3
[ 1 , 0 ]
Processing [level,i]= [ 1 , 0 ] Strategy = 3
SpairAndReduction:
[ p and bases , [ [ 0 , 2 ] , [ -9*y^2*Dy , 2*x ] ] , [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , %[null] ] ]
[ level= , 1 ]
[ tower2= , [ [ ] ] ]
[ 9*y^2*Dy , 2*es^2*x ]
[gi, gj] = [ -2*x*Dx-3*y*Dy+h^2 , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
1
Reduce the element -27*y^3*Dy^2+6*x*y*Dx*h^2-18*y^2*Dy*h^2+8*x^3*Dy*h
by [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , %[null] ]
result is [ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h , -1 , [ -3*y*h^2 , 0 , 0 , 0 ] ]
vdegree of the original = -1
vdegree of the remainder = -1
[ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h , [ -9*y^2*Dy-3*y*h^2 , 0 , -2*x , 0 ] , 0 , 3 , -1 , -1 ]
[ 1 , 2 ]
Processing [level,i]= [ 1 , 2 ] Strategy = 3
SpairAndReduction:
[ p and bases , [ [ 1 , 2 ] , [ -3*y*Dy , Dx ] ] , [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ] ]
[ level= , 1 ]
[ tower2= , [ [ ] ] ]
[ 3*es*y*Dy , es^2*Dx ]
[gi, gj] = [ -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
1
Reduce the element -6*y*Dx^2*h^2+4*x^2*Dx*Dy*h+6*x*y*Dy^2*h+8*x*Dy*h^3
by [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
result is [ 0 , 1 , [ 2*x*Dy*h , -2*h^2 , 0 , 0 ] ]
vdegree of the original = 1
vdegree of the remainder = %[null]
[ 0 , [ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ] , 2 , -1 , 1 , %[null] ]
[ 2 , 0 ]
Processing [level,i]= [ 2 , 0 ] Strategy = 3
SpairAndReduction:
[ p and bases , [ [ 0 , 3 ] , [ -Dx , 3*y*Dy ] ] , [ [ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ] , %[null] , [ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ] , [ 3*y*Dx , -2*x , 1 , 0 ] ] ]
[ level= , 2 ]
[ tower2= , [ [ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ] ] ]
[ Dx , -3*es^3*y*Dy ]
[gi, gj] = [ 9*y^2*Dy+2*es^2*x+es^3+3*y*h^2 , 3*y*Dx-2*es*x+es^2 ]
1
Reduce the element 6*es*x*y*Dy+2*es^2*x*Dx-3*es^2*y*Dy+es^3*Dx-6*y*Dx*h^2+2*es^2*h^2
by [ [ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ] , %[null] , [ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ] , [ 3*y*Dx , -2*x , 1 , 0 ] ]
result is [ -3*es^2*y*Dy+es^3*Dx+4*es^2*h^2-4*x^2*Dy*h , 1 , [ 0 , 0 , -2*x , 2*h^2 ] ]
vdegree of the original = 0
vdegree of the remainder = 0
[ -3*es^2*y*Dy+es^3*Dx+4*es^2*h^2-4*x^2*Dy*h , [ Dx , 0 , -2*x , -3*y*Dy+2*h^2 ] , 0 , 1 , 0 , 0 ]
[ 0 , 3 ]
Processing [level,i]= [ 0 , 3 ] Strategy = 4
[ 1 , 1 ]
Processing [level,i]= [ 1 , 1 ] Strategy = 4
Betti numbers are ------
[ 2 , 1 , 0 ]
[seq,level,q]=[ 3 , 1 , 1 ]
[ level, q = , 1 , 1 ]
bases=
[
[ -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
]
dr=
[ Dx , -1 , -2*x , -3*y*Dy+2*h^2 ]
newbases=
[
[ 0 , 0 , 0 , 0 ]
]
[seq,level,q]=[ 2 , 0 , 3 ]
[ level, q = , 0 , 3 ]
bases=
[
[ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
[ -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
[ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
[ 3*y*Dx , -2*x , 1 , 0 ]
]
dr=
[ -9*y^2*Dy-3*y*h^2 , 0 , -2*x , -1 ]
newbases=
[
[ 0 , 0 , 0 , 0 ]
[ -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , 0 , -2*x*Dx-3*y*Dy+2*h^2 , 0 ]
[ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
[ 3*y*Dx , -2*x , 1 , 0 ]
]
[seq,level,q]=[ 1 , 0 , 2 ]
[ level, q = , 0 , 2 ]
bases=
[
[ 0 , 0 , 0 , 0 ]
[ -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , 0 , -2*x*Dx-3*y*Dy+2*h^2 , 0 ]
[ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
[ 3*y*Dx , -2*x , 1 , 0 ]
]
dr=
[ -3*y*Dx , 2*x , -1 , 0 ]
newbases=
[
[ 0 , 0 , 0 , 0 ]
[ 6*x*y*Dx^2-4*x^2*Dy*h , -4*x^2*Dx-6*x*y*Dy , 0 , 0 ]
[ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy , 0 , 0 ]
[ 0 , 0 , 0 , 0 ]
]
[ level= , 0 ]
[
[ -2*x*Dx-3*y*Dy+h^2 ]
[ -3*y*Dx^2+2*x*Dy*h ]
]
[
[ -2*x*Dx-3*y*Dy+h^2 ]
[ -3*y*Dx^2+2*x*Dy*h ]
]
[ level= , 1 ]
[
[ 0 , 0 , 0 , 0 ]
[ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy , 0 , 0 ]
[ 0 , 0 , 0 , 0 ]
]
[
[ 0 , 0 ]
[ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
[ 0 , 0 ]
]
[ level= , 2 ]
[
[ 0 , 0 , 0 , 0 ]
]
[
[ 0 , 0 , 0 ]
]
------------ Note -----------------------------
To get shift vectors, use Reparse and SgetShifts(resmat,w)
To get initial of the complex, use Reparse and Sinit_w(resmat,w)
0: minimal resolution, 3: Schreyer resolution
------------ Resolution Summary --------------
Betti numbers : [ 2 , 1 ]
Betti numbers of the Schreyer frame: [ 4 , 4 , 1 ]
-----------------------------------------------
In(4)=sm1_pmat(a);
[
[
[
[ -2*x*Dx-3*y*Dy+h^2 ]
[ -3*y*Dx^2+2*x*Dy*h ]
]
[
[ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
]
]
[
[
[ -2*x*Dx-3*y*Dy+h^2 ]
[ -3*y*Dx^2+2*x*Dy*h ]
]
[
[ 0 , 0 ]
[ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
[ 0 , 0 ]
]
[
[ 0 , 0 , 0 ]
]
]
[
[
[
[ -2*x*Dx-3*y*Dy+h^2 ]
[ -3*y*Dx^2+2*x*Dy*h ]
[ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
[ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
]
[
[ 0 , 0 , 0 , 0 ]
[ 6*x*y*Dx^2-4*x^2*Dy*h , -4*x^2*Dx-6*x*y*Dy , 0 , 0 ]
[ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy , 0 , 0 ]
[ 0 , 0 , 0 , 0 ]
]
[
[ 0 , 0 , 0 , 0 ]
]
]
[
[ 0 , 0 , 1 , 2 ]
[ 0 , 3 , 0 , 0 ]
[ 0 ]
]
[
[ %[null] , %[null] , [ -3*y*Dx , 2*x , -1 , 0 ] , [ -9*y^2*Dy-3*y*h^2 , 0 , -2*x , -1 ] ]
[ %[null] , [ Dx , -1 , -2*x , -3*y*Dy+2*h^2 ] , %[null] , %[null] ]
[ %[null] ]
]
[ 1 , 4 , 4 , 1 ]
[
[ 0 , 0 , 1 , 2 ]
[ 0 , 3 , %[null] , 0 ]
[ 0 ]
]
]
[
[
[ -2*x*Dx-3*y*Dy+h^2 ]
[ -3*y*Dx^2+2*x*Dy*h ]
[ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
[ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
]
[
[ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
[ -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
[ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
[ 3*y*Dx , -2*x , 1 , 0 ]
]
[
[ -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
]
]
[
[
[
[ -9*y^2*Dy , 0 , 2*x , 0 ]
[ 0 , 0 , -3*y*Dy , Dx ]
[ 0 , -3*y*Dy , Dx , 0 ]
[ -3*y*Dx , 2*x , 0 , 0 ]
]
[
[ -Dx , 0 , 0 , 3*y*Dy ]
]
[ ]
]
[
[ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
[ -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
[ -Dx ]
]
[
[ ]
[
[
[ 0 , 2 ]
[ -9*y^2*Dy , 2*x ]
]
[
[ 2 , 3 ]
[ -3*y*Dy , Dx ]
]
[
[ 1 , 2 ]
[ -3*y*Dy , Dx ]
]
[
[ 0 , 1 ]
[ -3*y*Dx , 2*x ]
]
]
[
[
[ 0 , 3 ]
[ -Dx , 3*y*Dy ]
]
]
[ ]
]
[
[ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , -9*y^2*Dx*Dy-3*y*Dx*h^2-4*x^2*Dy*h , -27*y^3*Dy^2-27*y^2*Dy*h^2+3*y*h^4+8*x^3*Dy*h ]
]
]
]
In(5)=![[BACK]](/icons/cvsweb/back.gif){kind=icon}
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