File: [local] / OpenXM / src / k097 / lib / minimal / minimal-test.k (download)
Revision 1.5, Thu Jun 15 07:38:35 2000 UTC (24 years ago) by takayama
Branch: MAIN
Changes since 1.4: +27 -3
lines
ScheckIfSchreyer(): it checks if grade (grading function)
is module1v (grading without vector component).
Note that ring_of_differential_operators switches grade to
module1.
Found a reason why the strategy fails.
It is because the schreyer skelton contains an element
of the form [1,f]. This bug will be fixed soon.
A list of todo (e.g., init<m> )
Succeeded to construct the V-minimal resolution for 1/(x^3+y^3+z^3).
The betti numbers are 4,5,2.
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/* $OpenXM: OpenXM/src/k097/lib/minimal/minimal-test.k,v 1.5 2000/06/15 07:38:35 takayama Exp $ */
load["minimal.k"];
def test5() {
local a,b,c,cc,v;
a = Sannfs3_laScala2("x^3-y^2*z^2");
b = a[0];
v = [x,y,z];
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
Println("-----------------------------------");
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
return(a);
}
def test6() {
local a,b,c,cc,v;
a = Sannfs3("x^3-y^2*z^2");
b = a[0];
v = [x,y,z];
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
Println("-------ker = im for minimal ?---------------------");
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
Println("------ ker=im for Schreyer ?------------------");
b = a[3];
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
return(a);
}
/* May 23, Tue */
def test7() {
local a,b,c,cc,v;
a = Sannfs3_laScala2("x^3-y^2*z^2");
b = a[0];
v = [x,y,z];
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
Println("-------ker = im for minimal ?---------------------");
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
Println("------ ker=im for Schreyer ?------------------");
b = a[3];
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
return(a);
}
def sm1_resol1(p) {
sm1(" p resol1 /FunctionValue set ");
}
def test8() {
local p,pp,ans,b,c,cc,ww,ww2;
f = "x^3-y^2*z^2";
p = Sannfs(f,"x,y,z");
ww = [["x",1,"y",1,"z",1,"Dx",1,"Dy",1,"Dz",1,"h",1],
["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
Sweyl("x,y,z",ww);
pp = Map(p,"Spoly");
/* return(pp); */
/* pp =
[y*Dy-z*Dz , -2*x*Dx-3*y*Dy+1 , 2*x*Dy*Dz^2-3*y*Dx^2 ,
2*x*Dy^2*Dz-3*z*Dx^2 , 2*x*z*Dz^3-3*y^2*Dx^2+4*x*Dz^2 ]
*/
ans = sm1_resol1([pp,"x,y,z",ww]);
/* Schreyer is in ans. */
v = [x,y,z];
b = ans;
Println("------ ker=im for Schreyer ?------------------");
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
return(ans);
}
/*
a = test8();
SisComplex(a):
*/
def test8a() {
local p,pp,ans,b,c,cc,ww, ans_all;
f = "x^3-y^2*z^2";
p = Sannfs(f,"x,y,z");
sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
ww = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
/* Removed "x",1, ... ===> It causes an error. I do not know the reason.*/
Sweyl("x,y,z",ww);
pp = Map(p,"Spoly");
/* return(pp); */
/* pp =
[y*Dy-z*Dz , -2*x*Dx-3*y*Dy+1 , 2*x*Dy*Dz^2-3*y*Dx^2 ,
2*x*Dy^2*Dz-3*z*Dx^2 , 2*x*z*Dz^3-3*y^2*Dx^2+4*x*Dz^2 ]
*/
ans_all = Sschreyer(pp);
ans = ans_all[0];
/* ans = sm1_resol1([pp,"x,y,z",ww]); */
/* Schreyer is in ans. */
v = [x,y,z];
b = ans;
Println("------ ker=im for Schreyer ?----- wrong method!!!-----------");
c = Skernel(b[0],v);
c = c[0];
sm1_pmat([c,b[1],v]);
cc = sm1_res_div(c,b[1],v);
sm1_pmat(sm1_gb(cc,v));
c = Skernel(b[1],v);
c = c[0];
cc = sm1_res_div(c,b[2],v);
sm1_pmat(sm1_gb(cc,v));
return(ans);
}
/* Comparing two constructions */
def test9() {
local p,pp,ans,b,c,cc,ww,ww2,ans_all,ans2;
f = "x^3-y^2*z^2";
p = Sannfs(f,"x,y,z");
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
Sweyl("x,y,z",ww2);
pp = Map(p,"Spoly");
ans = sm1_resol1([pp,"x,y,z",ww2]);
f = "x^3-y^2*z^2";
p = Sannfs(f,"x,y,z");
sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
ww = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Sweyl("x,y,z",ww);
pp = Map(p,"Spoly");
ans_all = Sschreyer(pp);
ans2 = ans_all[0];
return([ans,ans2]);
}
/* Check if the complex by Sschreyer() is exact or not in our example? */
def test10() {
local p,pp,ans,b,c,cc,ww,ww2,ans_all,ans2, r;
f = "x^3-y^2*z^2";
p = Sannfs(f,"x,y,z");
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
Sweyl("x,y,z",ww2);
pp = Map(p,"Spoly");
ans = sm1_resol1([pp,"x,y,z",ww2]);
f = "x^3-y^2*z^2";
p = Sannfs(f,"x,y,z");
sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
ww = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Sweyl("x,y,z",ww);
pp = Map(p,"Spoly");
ans_all = Sschreyer(pp); /* Schreyer by LaScala-Stillman */
ans2 = ans_all[0];
sm1(" /gb.verbose 1 def ");
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Sweyl("x,y,z",ww2);
ans2 = ReParse(ans2);
r= IsExact_h(ans2,[x,y,z]);
Print(r);
return([r,[ans,ans2]]);
}
def test11() {
local a;
a = test_ann3("x^3-y^2*z^2");
return(a);
}
/* f should be a string. */
/* a=test_ann3("x^3+y^3+z^3");
It returns the following resolution in 1.5 hours. June 14, 2000.
[
[
[ x*Dx+y*Dy+z*Dz-3*h^2 ]
[ -z*Dy^2+y*Dz^2 ]
[ -z*Dx^2+x*Dz^2 ]
[ -y*Dx^2+x*Dy^2 ]
]
[
[ 0 , -x , y , -z ]
[ z*Dx^2-x*Dz^2 , x*Dy , x*Dx+z*Dz-3*h^2 , z*Dy ]
[ y*Dx^2-x*Dy^2 , -x*Dz , y*Dz , x*Dx+y*Dy-3*h^2 ]
[ 0 , Dx^2 , -Dy^2 , Dz^2 ]
[ z*Dy^2-y*Dz^2 , x*Dx+y*Dy+z*Dz-2*h^2 , 0 , 0 ]
]
[
[ -x*Dx+3*h^2 , y , -z , 0 , -x ]
[ Dy^3+Dz^3 , Dy^2 , -Dz^2 , x*Dx+y*Dy+z*Dz , -Dx^2 ]
]
]
*/
def test_ann3(f) {
local a,v,ww2,ans2;
a = Sannfs3_laScala2(f);
ans2 = a[0];
v = [x,y,z];
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Sweyl("x,y,z",ww2);
ans2 = ReParse(ans2);
r= IsExact_h(ans2,[x,y,z]);
Println(r);
return([r,ans2]);
}
def test11a() {
local a,v,ww2,ans2;
/* constructed by test11.
ans2 =
[[[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]] ,
[[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] ,
[0 , 0 , 0 , 0] ,
[2*x*Dy*Dz , 0 , z , -y] ,
[0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0]] ,
[[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[-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] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[3*y*z , z , y , -2*x*Dy*Dz , -3*z*Dy , 2*x*Dx , 2*x*z , -2*x*y , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0]] ,
[[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0] ,
[0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0]]]
*/
ans2 =
[[[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]] ,
[[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*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]]];
sm1_pmat( ans2[1]*ans2[0] );
sm1_pmat( ans2[2]*ans2[1] );
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Sweyl("x,y,z",ww2);
ans2 = ReParse(ans2);
r= IsExact_h(ans2,[x,y,z]);
Println(r);
return([r,ans2]);
}
def test12() {
local a,v,ww2,ans2;
a = Sannfs3("x^3-y^2*z^2");
ans2 = a[0];
v = [x,y,z];
ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Sweyl("x,y,z",ww2);
ans2 = ReParse(ans2); /* DO NOT FORGET! */
r= IsExact_h(ans2,[x,y,z]);
Println(r);
Println("It may stop by non-exact statement. The code of Sminimal_v (non-LaScala-Stillman contains bugs.");
return([r,ans2]);
}
def test13() {
Println("test13 try to construct a minimal free resolution");
Println("of a GKZ system [[1,2]]. 6/12, 2000.");
ans2 = GKZ([[1,2]],[0]);
/* Be careful!! It resets the grade to module1, not module1v */
ww2 = [["x1",-1,"x2",-1,"Dx1",1,"Dx2",1]];
Sweyl("x1,x2",ww2);
ans2 = ReParse(ans2[0]);
Println(ans2);
return(Sminimal(ans2));
}
def test14() {
Println("test14 try to construct a minimal free resolution");
Println("of a GKZ system [[1,2,3]]. 6/12, 2000.");
ans2 = GKZ([[1,2,3]],[0]); /* It stops by the strategy error. */
ww2 = [["x1",-1,"x2",-1,"x3",-1,"Dx1",1,"Dx2",1,"Dx3",1]];
Sweyl("x1,x2,x3",ww2);
ans2 = ReParse(ans2[0]);
return(Sminimal(ans2));
}
def test14a() {
Println("test14a try to construct a minimal free resolution");
Println("of a GKZ system [[1,2,3]]. 6/12, 2000.");
Println("Without automatic homogenization.");
ww2 = [["x1",-1,"x2",-1,"x3",-1,"Dx1",1,"Dx2",1,"Dx3",1]];
Sweyl("x1,x2,x3",ww2);
ans2 = [x1*Dx1+2*x2*Dx2+3*x3*Dx3 , Dx1^2-Dx2*h , -Dx1*Dx2+Dx3*h ,
Dx2^2-Dx1*Dx3 ];
ans2 = ReParse(ans2);
return(Sminimal(ans2,"homogenized"));
}
def test15() {
Println("test15 try to construct a minimal free resolution");
Println("of a GKZ system [[1,2,3]] by the order filt. 6/12, 2000.");
ww2 = [["Dx1",1,"Dx2",1,"Dx3",1]];
Sweyl("x1,x2,x3",ww2);
ans2 = GKZ([[1,2,3]],[0]);
ans2 = ReParse(ans2[0]);
return(Sminimal(ans2));
}
def test15b() {
Println("test15b try to construct a minimal free resolution");
Println("of toric [[1,2,3]] by the order filt. 6/12, 2000.");
ww2 = [["Dx1",1,"Dx2",1,"Dx3",1]];
Sweyl("x1,x2,x3",ww2);
ans2 = [Dx1^2-Dx2*h , -Dx1*Dx2+Dx3*h , Dx2^2-Dx1*Dx3 ];
ans2 = ReParse(ans2);
return(Sminimal(ans2,"homogenized"));
}
def test16() {
Println("test16 try to construct a minimal free resolution");
Println("of a GKZ system [[1,2,3,5]] by the order filt. 6/12, 2000.");
ww2 = [["Dx1",1,"Dx2",1,"Dx3",1,"Dx4",1]];
Sweyl("x1,x2,x3,x4",ww2);
ans2 = GKZ([[1,2,3,5]],[0]);
ans2 = ReParse(ans2[0]);
return(Sminimal(ans2));
}
def test16b() {
Println("test16b try to construct a minimal free resolution");
Println("of a toric [[1,2,3,5]] by the order filt. 6/12, 2000.");
ww2 = [["Dx1",1,"Dx2",1,"Dx3",1,"Dx4",1]];
Sweyl("x1,x2,x3,x4",ww2);
ans2 = GKZ([[1,2,3,5]],[0]);
ans3 = Rest(ans2[0]);
ans3 = ReParse(ans3);
Println("Toric variety:");
Println(ans3);
return(Sminimal(ans3));
}