| version 1.3, 2000/06/08 08:37:53 |
version 1.12, 2000/08/09 03:45:27 |
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| $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.2 2000/05/24 15:24:54 takayama Exp $ |
$OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.11 2000/08/02 05:14:30 takayama Exp $ |
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| SpairAndReduction() : |
SpairAndReduction() : |
| $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B. |
$BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B. |
| Line 77 test8() $B$G(B sm1 $B$G=q$$$?J}$N(B Schreyer $B$r |
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| Line 77 test8() $B$G(B sm1 $B$G=q$$$?J}$N(B Schreyer $B$r |
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| kernel = image |
kernel = image |
| $B$H$J$C$F$$$k$N$G0J8e$3$N(B option $B$O(B 1 $B$N$^$^;H$&$3$H$H$9$k(B. |
$B$H$J$C$F$$$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. |
$BMW$9$k$K(B k0 $B$N%3!<%I$,$I$&$d$i$*$+$7$$$i$7$$(B. |
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==> |
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6/8 $B$N%N!<%H$h$j(B. |
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syzygy $B$r(B homogenization $B$r2p$7$F7W;;$9$k$N$OLdBj$"$j(B. |
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--> usage of isExact |
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$BMW$9$k$K(B kernel = image $B$N%3!<%I$bJQ(B. Homogenized $B$N$^$^$d$kI,MW$"$j(B. |
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| ----------------------------------- |
----------------------------------- |
| June 8, 2000 (Thu), 9:10 (Spain local time) |
June 8, 2000 (Thu), 9:10 (Spain local time) |
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| LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, schreyer resol $B$,(B exact $B$+(B |
LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, schreyer resol $B$,(B exact $B$+(B |
| $BD4$Y$k(B. |
$BD4$Y$k(B. |
| $BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B. |
$BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B. |
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==> OK. IsExact_h $B$G$7$i$Y$k(B. (IsExact $B$O$@$a$h(B) |
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June 8, 2000 (Thu), 19:35 |
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load["minimal-test.k"];; |
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test11(); |
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LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, minimal resol $B$,(B exact $B$+(B |
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$BD4$Y$k(B. |
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$BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B. |
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SwhereInTower $B$r;H$&$H$-$O(B, |
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SsetTower() $B$G(B gbList $B$rJQ99$7$J$$$H$$$1$J$$(B. |
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$B$b$A$m$s;HMQ$7$?$i(B, $B$=$l$rLa$9$3$H(B. |
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SpairAndReduction, SpairAndReduction2 $B$G(B, |
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SsetTower(StowerOf(tower,level)); |
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pos = SwhereInTower(syzHead,tower[level]); |
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SsetTower(StowerOf(tower,level-1)); |
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pos2 = SwhereInTower(tmp[0],tower[level-1]); |
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$B$H(B, SwhereInTower $B$NA0$K(B setTower $B$r$/$o$($?(B. |
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( $B0c$&%l%Y%k$G$NHf3S$N$?$a(B.) |
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IsExact_h $B$O(B, 0 $B%Y%/%H%k$r4^$`>l9g(B, $B$?$@$7$/F0:n$7$J$$$h$&$@(B. |
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test11(). |
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test11a() $B$G(B, 0 $B%Y%/%H%k$r<j$G=|$$$?9TNs$N(B exactness $B$r%A%'%C%/(B. ==> OK. |
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--------------------------------- |
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June 9, 6:20 |
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SpairAndReduction |
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$B$H(B |
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SpairAndReduction2 |
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$B$N0c$$(B. |
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SpairAndReduction : SlaScala (LaScala-Stillman's algorithm $B$G;H$&(B) |
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SpairAndReduction2 : Sschreyer (schreyer algorithm $B$G;H$&(B, laScala $B$O$J$7(B.) |
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0 $B$r<+F0$G=|$/%3!<%I$r=q$3$&(B. |
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SpruneZeroRow() $B$r(B Sminimal() $B$K2C$($?(B. |
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test11() $B$b@5$7$/F0:n$9$k$O$:(B. |
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IsExact_h $B$O(B schreyer $B$r(B off $B$7$F(B, ReParse $B$7$F$+$i(B, |
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$B8F$S=P$9$3$H(B. |
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#ifdef TOTAL_STRATEGY |
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return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1)); |
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#endif |
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/* Strategy must be compatible with ordering. */ |
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/* Weight vector must be non-negative, too. */ |
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/* See Sdegree, SgenerateTable, reductionTable. */ |
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wd = Sord_w(f,ww); |
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return(wd+Sdegree(tower[level-2,i],tower,level-1)); |
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TOTAL_STRATEGY $B$rMQ$$$kI,MW$,$"$k$N$G$O(B?? |
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Example 1: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]); |
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v=[[2*x*Dx + 3*y*Dy+6, 0], |
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[3*x^2*Dy + 2*y*Dx, 0], |
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[0, x^2+y^2], |
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[0, x*y]]; |
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a=Sminimal(v); |
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strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B? |
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a=test_ann3("x^3+y^3+z^3); $B$O;~4V$,$+$+$j$=$&(B. |
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a=test_ann3("x^3+y^3"); OK. |
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a=test_ann3("x^2+y^2+z"); OK. |
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$B>e$N(B example 1 $B$N%(%i!<(B $B$N8+J}(B: |
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Processing [ 1 , 3 ] Strategy = 2 |
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1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B. |
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In(7)=reductionTable: |
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[[ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ] |
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-- $B$3$l(B. |
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SpairAndReduction: |
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[ 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] ] ] |
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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. |
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[ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B. |
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( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. ) |
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[ level= , 1 ] |
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[ tower2= , [ [ ] ] ] ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. ) |
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[ y*h , -es^3*x ] |
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[gi, gj] = [ 2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ] |
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1 |
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Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy |
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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] ] |
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result is [ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [ 0 , 0 , 0 , 0 , 0 , 0 ] ] |
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vdegree of the original = -1 |
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vdegree of the remainder = -1 |
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[ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [ y*h , 0 , 0 , -x , 0 , 0 ] , 3 , 5 , -1 , -1 ] |
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In(11)=freeRes: |
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[ [ 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] ] ] |
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$B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B |
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$B$i$l$?(B. |
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$B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B. |
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i = SnextI(reductionTable_tmp,strategy,redundantTable, |
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skel,level,freeRes); |
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In(22)=reductionTable: |
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[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ] |
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$B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B, |
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In(25)=skel[2]: |
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[ [ [ 0 , 2 ] , [ 1 , -y^2 ] ] ] |
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$B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair. |
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$B$7$+$7(B, |
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In(26)=bases: |
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[ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ] |
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$B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B. |
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reductionTable_tmp=[ 2 ] |
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See also reductionTable, strategy, level,i |
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ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations. |
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--- Engine error or interrupt : In function : Error of class PrimitiveObject |
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Type in Cleards() to exit the debug mode and Where() to see the stack trace. |
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In(7)=reductionTable: |
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[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ] |
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In(8)=strategy: |
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2 |
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In(9)=level: |
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2 |
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RemoveRedundantInSchreyerSkelton = 0 |
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$B$H$7$F$bF1$8%(%i!<(B. |
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------------------------------------------------- |
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test_ann3("x*y+y*z+z*x"); OK. |
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6/9 (Fri) |
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Sminimal $B$N<BAu$KAjJQ$o$i$:6lO+$7$F$^$9(B. |
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Sevilla $B$G$$$m$$$m$HD>$7$?7k2L(B, |
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Sminimal $B$O$&$^$/$&$4$1$P@5$7$$Ez$($r$@$7$F$k$_$?$$$G$9$,(B |
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(D<h> : homogenized Weyl $B$G(B ker = im $B$r(B check $B$7$F$k(B, |
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V-adapted (strict) $B$+$I$&$+$N(B check routing $B$O$^$@=q$$$F$J$$(B), |
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strategy $B$,$&$^$/$&$4$+$J$/$F$H$^$k>l9g$b$"$j$^$9(B |
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( strategy = 2 $B$N(B sp $B$r7W;;$9$k$N$K(B, strategy 3 $B$N(B $B85$rI,MW$H(B |
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$B$7$?$j$9$k>l9g$"$j(B). |
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strategy $B$O(B |
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def Sdegree(f,tower,level) { |
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local i,ww, wd; |
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/* extern WeightOfSweyl; */ |
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ww = WeightOfSweyl; |
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f = Init(f); |
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if (level <= 1) return(StotalDegree(f)); |
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i = Degree(f,es); |
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return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1)); |
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} |
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$B$rMQ$$$F(B, |
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ans_at_each_floor[j] = Sdegree(tower[i,j],tower,i+1)-(i+1) |
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$B$G7W;;$7$F$^$9(B. |
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$B$$$/$D$+=PNO$r$D$1$F$*$-$^$9$N$G(B, $B8!F$(B!!! |
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$BNc(B 1: |
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load["minimal-test.k"];; |
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a=test_ann3("x^3-y^2*z^2"); $B0z?t$N(B annihilating ideal $B$N(B laplace $BJQ49$N(B |
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homogenization $B$N(B resolution. |
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weight vector $B$O(B (-1,-1,-1,1,1,1) |
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In(4)=sm1_pmat(a[1]); |
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[ |
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[ 0 $B<!(B |
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[ y*Dy-z*Dz ] |
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[ -2*x*Dx-3*z*Dz+h^2 ] |
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[ 2*x*Dy*Dz^2-3*y*Dx^2*h ] |
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[ 2*x*Dy^2*Dz-3*z*Dx^2*h ] |
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] |
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[ 1 $B<!(B |
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[ 3*Dx^2*h , 0 , Dy , -Dz ] |
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[ 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 ] |
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[ 0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ] |
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[ 2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ] |
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[ 2*x*Dy*Dz , 0 , z , -y ] |
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] |
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[ 2 $B<!(B |
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[ -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy*Dz ] |
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[ 3*y*z , z , y , -2*x*Dy*Dz , 2*x*Dx ] |
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] |
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] |
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In(5)= |
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$BNc(B 2: |
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load["minimal-test.k"];; |
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a=test_ann3("x*y+y*z+z*x"); |
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In(6)=sm1_pmat(a[1]); |
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[ |
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[ 0 $B<!(B |
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[ 2*x*Dx+x*Dz-y*Dz+z*Dz+h^2 ] |
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[ -2*y*Dy+x*Dz-y*Dz-z*Dz-h^2 ] |
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[ -2*x*Dy+2*z*Dy+x*Dz-y*Dz+3*z*Dz+h^2 ] |
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[ -2*y*Dx+2*z*Dx-x*Dz+y*Dz+3*z*Dz+h^2 ] |
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] |
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[ 1 $B<!(B |
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[ y-z , x-z , -y , x ] |
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[ 2*Dy-2*Dz , 2*Dx-2*Dz , 2*Dx+2*Dz , -2*Dy-2*Dz ] |
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[ 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 ] |
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[ 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 ] |
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[ -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 ] |
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] |
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[ 2 $B<!(B |
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[ -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 ] |
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[ -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 ] |
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] |
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] |
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In(7)= |
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$BNc(B 3: $B$&$^$/9T$+$J$$Nc(B: |
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Example 1: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]); |
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v=[[2*x*Dx + 3*y*Dy+6, 0], |
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[3*x^2*Dy + 2*y*Dx, 0], |
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[0, x^2+y^2], |
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[0, x*y]]; |
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a=Sminimal(v); |
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strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B? |
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Negative weight vector $B$r;H$o$J$$$H$-$A$s$HF0$-$^$9(B. |
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DEBUG $B=PNO(B: |
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rf= [ |
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[ |
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[ Schreyer frame. |
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[ 0 , y^3 , 0 , 0 , -x^2 , 0 ] |
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[ 0 , 0 , y^2 , 0 , -x , 0 ] |
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[ 0 , y , -x , 0 , 0 , 0 ] |
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[ y*h , 0 , 0 , -x , 0 , 0 ] |
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[ 0 , 0 , 0 , 3*y*Dy , 0 , -2*Dx ] |
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] |
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[ |
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[ 1 , 0 , -y^2 , 0 , 0 ] |
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] |
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[ ] |
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] |
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[ |
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[ 2*x*Dx , e_*x^2 , e_*x*y , 2*y*Dx*h , e_*y^3 , 3*y^2*Dy*h ] |
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[ es*y^3 , es^2*y^2 , es*y , y*h , 3*es^3*y*Dy ] |
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[ 1 ] |
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] |
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[ |
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[ ] |
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[ |
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[ |
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[ 1 , 4 ] |
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[ y^3 , -x^2 ] |
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] |
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[ |
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[ 2 , 4 ] |
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[ y^2 , -x ] |
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] |
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[ |
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[ 1 , 2 ] |
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[ y , -x ] |
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] |
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[ |
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[ 0 , 3 ] |
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[ y*h , -x ] |
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] |
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[ |
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[ 3 , 5 ] |
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[ 3*y*Dy , -2*Dx ] |
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] |
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] |
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[ |
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[ |
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[ 0 , 2 ] |
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[ 1 , -y^2 ] |
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] |
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] |
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[ ] |
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] |
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[ resolution $B$9$Y$-(B $BItJ,2C72(B e_ $B$O(B $B%Y%/%H%k@.J,$N%^!<%/(B. |
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[ 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 ] |
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] |
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] |
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$BN,(B |
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Processing [ 1 , 3 ] Strategy = 2 |
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1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B. |
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In(7)=reductionTable: |
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[[ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ] |
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-- $B$3$l(B. |
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SpairAndReduction: |
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[ 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] ] ] |
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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. |
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[ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B. |
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( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. ) |
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[ level= , 1 ] |
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[ tower2= , [ [ ] ] ] ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. ) |
| |
[ y*h , -es^3*x ] |
| |
[gi, gj] = [ 2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ] |
| |
1 |
| |
Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy |
| |
by [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ] |
| |
result is [ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [ 0 , 0 , 0 , 0 , 0 , 0 ] ] |
| |
vdegree of the original = -1 |
| |
vdegree of the remainder = -1 |
| |
[ 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [ y*h , 0 , 0 , -x , 0 , 0 ] , 3 , 5 , -1 , -1 ] |
| |
|
| |
In(11)=freeRes: |
| |
[ [ 2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ] , [ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ] , [ %[null] ] ] |
| |
$B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B |
| |
$B$i$l$?(B. |
| |
|
| |
$B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B. |
| |
i = SnextI(reductionTable_tmp,strategy,redundantTable, |
| |
skel,level,freeRes); |
| |
In(22)=reductionTable: |
| |
[ [ 1 , 1 , 1 , 2 , 2 , 3 ] , [ 3 , 2 , 1 , 2 , 3 ] , [ 2 ] ] |
| |
$B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B, |
| |
In(25)=skel[2]: |
| |
[ [ [ 0 , 2 ] , [ 1 , -y^2 ] ] ] |
| |
$B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair. |
| |
$B$7$+$7(B, |
| |
In(26)=bases: |
| |
[ %[null] , [ 0 , 0 , y^2 , 0 , -x , 0 ] , [ 0 , -y , x , 0 , 1 , 0 ] , [ -y*h , 0 , 0 , x , 0 , 1 ] , %[null] ] |
| |
$B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$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)= |
| |
|
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