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D.15.17.50 hilbertSeries
Procedure from library multigrading.lib (see multigrading_lib).
- Usage:
- hilbertSeries(I); I is poly/vector/ideal/module
- Purpose:
- computes the multigraded Hilbert Series of I
- Note:
- input must have multigraded-homogeneous generators.
Multigrading should be positive.
- Returns:
- a ring in variables t_(i), s_(i), with polynomials
numerator1 and denominator1 and mutually prime numerator2
and denominator2, quotients of which give the series.
Example:
| | LIB "multigrading.lib";
ring r = 0,(x,y,z,w),dp;
intmat g[2][4]=
1,1,1,1,
0,1,3,4;
setBaseMultigrading(g);
module M = ideal(xw-yz, x2z-y3, xz2-y2w, yw2-z3);
intmat V[2][1]=
1,
0;
M = setModuleGrading(M, V);
def h = hilbertSeries(M); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
factorize(numerator2);
==> [1]:
==> _[1]=-1
==> _[2]=t_(1)
==> _[3]=t_(1)^3*t_(2)^6-t_(1)^2*t_(2)^6-t_(1)^2*t_(2)^2-t_(1)*t_(2)^3-t_(\
1)*t_(2)-1
==> [2]:
==> 1,1,1
factorize(denominator2);
==> [1]:
==> _[1]=1
==> _[2]=t_(1)-1
==> _[3]=t_(1)*t_(2)^4-1
==> [2]:
==> 1,1,1
kill g, h; setring r;
intmat g[2][4]=
1,2,3,4,
0,0,5,8;
setBaseMultigrading(g);
ideal I = x^2, y, z^3;
I = std(I);
list L = multiDegResolution(I, 0, 1);
for( int j = 1; j<=size(L); j++)
{
"----------------------------------- ", j, " -----------------------------";
L[j];
"Module Multigrading: "; print( getModuleGrading(L[j]) );
"Multigrading: "; print(multiDeg(L[j]));
}
==> ----------------------------------- 1 -----------------------------
==> _[1]=y
==> _[2]=x2
==> _[3]=z3
==> Module Multigrading:
==> 0
==> 0
==> Multigrading:
==> 2 2 9
==> 0 0 15
==> ----------------------------------- 2 -----------------------------
==> _[1]=-x2*gen(1)+y*gen(2)
==> _[2]=-z3*gen(1)+y*gen(3)
==> _[3]=-z3*gen(2)+x2*gen(3)
==> Module Multigrading:
==> 2 2 9
==> 0 0 15
==> Multigrading:
==> 4 11 11
==> 0 15 15
==> ----------------------------------- 3 -----------------------------
==> _[1]=z3*gen(1)-x2*gen(2)+y*gen(3)
==> Module Multigrading:
==> 4 11 11
==> 0 15 15
==> Multigrading:
==> 13
==> 15
multiDeg(I);
==> 2,2,9,
==> 0,0,15
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
factorize(numerator2);
==> [1]:
==> _[1]=-1
==> _[2]=t_(1)+1
==> _[3]=t_(1)^6*t_(2)^10+t_(1)^3*t_(2)^5+1
==> [2]:
==> 1,1,1
factorize(denominator2);
==> [1]:
==> _[1]=1
==> _[2]=t_(1)*t_(2)^2-1
==> _[3]=t_(1)*t_(2)^2+1
==> _[4]=t_(1)^2*t_(2)^4+1
==> [2]:
==> 1,1,1,1
kill r, h, g, V;
////////////////////////////////////////////////
ring R = 0,(x,y,z),dp;
intmat W[2][3] =
1,1, 1,
0,0,-1;
setBaseMultigrading(W);
ideal I = x3y,yz2,y2z,z4;
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
factorize(numerator2);
==> [1]:
==> _[1]=1
==> _[2]=-t_(1)^5*s_(2)+t_(1)^3*s_(2)^3+t_(1)^2*s_(2)^2+t_(1)^3+t_(1)^2*s_\
(2)+t_(1)^2+t_(1)*s_(2)+t_(1)+1
==> [2]:
==> 1,1
factorize(denominator2);
==> [1]:
==> _[1]=-1
==> _[2]=t_(1)-1
==> [2]:
==> 1,1
kill R, W, h;
////////////////////////////////////////////////
ring R = 0,(x,y,z,a,b,c),dp;
intmat W[2][6] =
1,1, 1,1,1,1,
0,0,-1,0,0,0;
setBaseMultigrading(W);
ideal I = x3y,yz2,y2z,z4;
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
factorize(numerator2);
==> [1]:
==> _[1]=1
==> _[2]=-t_(1)^5*s_(2)+t_(1)^3*s_(2)^3+t_(1)^2*s_(2)^2+t_(1)^3+t_(1)^2*s_\
(2)+t_(1)^2+t_(1)*s_(2)+t_(1)+1
==> [2]:
==> 1,1
factorize(denominator2);
==> [1]:
==> _[1]=1
==> _[2]=t_(1)-1
==> [2]:
==> 1,4
kill R, W, h;
////////////////////////////////////////////////
// This is example 5.3.9. from Robbianos book.
ring R = 0,(x,y,z,w),dp;
intmat W[1][4] =
1,1, 1,1;
setBaseMultigrading(W);
ideal I = z3,y3zw2,x2y4w2xyz2;
hilb(std(I));
==> // 1 t^0
==> // -1 t^3
==> // -1 t^6
==> // 1 t^8
==>
==> // 1 t^0
==> // 1 t^1
==> // 1 t^2
==> // -1 t^6
==> // -1 t^7
==> // dimension (proj.) = 2
==> // degree (proj.) = 1
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1;
==> t_(1)^8-t_(1)^6-t_(1)^3+1
denominator1;
==> t_(1)^4-4*t_(1)^3+6*t_(1)^2-4*t_(1)+1
factorize(numerator2);
==> [1]:
==> _[1]=1
==> _[2]=t_(1)^7+t_(1)^6-t_(1)^2-t_(1)-1
==> [2]:
==> 1,1
factorize(denominator2);
==> [1]:
==> _[1]=1
==> _[2]=t_(1)-1
==> [2]:
==> 1,3
kill h;
////////////////////////////////////////////////
setring R;
ideal I2 = x2,y2,z2; I2;
==> I2[1]=x2
==> I2[2]=y2
==> I2[3]=z2
hilb(std(I2));
==> // 1 t^0
==> // -3 t^2
==> // 3 t^4
==> // -1 t^6
==>
==> // 1 t^0
==> // 3 t^1
==> // 3 t^2
==> // 1 t^3
==> // dimension (proj.) = 0
==> // degree (proj.) = 8
def h = hilbertSeries(I2); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1;
==> -t_(1)^6+3*t_(1)^4-3*t_(1)^2+1
denominator1;
==> t_(1)^4-4*t_(1)^3+6*t_(1)^2-4*t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
W = 2,2,2,2;
setBaseMultigrading(W);
getVariableWeights();
==> 2,2,2,2
intvec w = 2,2,2,2;
hilb(std(I2), 1, w);
==> 1,0,0,0,-3,0,0,0,3,0,0,0,-1,0
kill w;
def h = hilbertSeries(I2); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -t_(1)^12+3*t_(1)^8-3*t_(1)^4+1
==> t_(1)^8-4*t_(1)^6+6*t_(1)^4-4*t_(1)^2+1
kill h;
kill R, W;
////////////////////////////////////////////////
ring R = 0,(x),dp;
intmat W[1][1] =
1;
setBaseMultigrading(W);
ideal I;
I = 1; I;
==> I[1]=1
hilb(std(I));
==>
==> // dimension (affine) = 0
==> // degree (affine) = 0
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> 0
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
I = x; I;
==> I[1]=x
hilb(std(I));
==> // 1 t^0
==> // -1 t^1
==>
==> // 1 t^0
==> // dimension (affine) = 0
==> // degree (affine) = 1
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -t_(1)+1
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
I = x^5; I;
==> I[1]=x5
hilb(std(I));
==> // 1 t^0
==> // -1 t^5
==>
==> // 1 t^0
==> // 1 t^1
==> // 1 t^2
==> // 1 t^3
==> // 1 t^4
==> // dimension (affine) = 0
==> // degree (affine) = 5
hilb(std(I), 1);
==> 1,0,0,0,0,-1,0
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -t_(1)^5+1
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
I = x^10; I;
==> I[1]=x10
hilb(std(I));
==> // 1 t^0
==> // -1 t^10
==>
==> // 1 t^0
==> // 1 t^1
==> // 1 t^2
==> // 1 t^3
==> // 1 t^4
==> // 1 t^5
==> // 1 t^6
==> // 1 t^7
==> // 1 t^8
==> // 1 t^9
==> // dimension (affine) = 0
==> // degree (affine) = 10
def h = hilbertSeries(I); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -t_(1)^10+1
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
module M = 1;
M = setModuleGrading(M, W);
hilb(std(M));
==>
==> // dimension (affine) = 0
==> // degree (affine) = 0
def h = hilbertSeries(M); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> 0
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
kill M; module M = x^5*gen(1);
// intmat V[1][3] = 0; // TODO: this would lead to a wrong result!!!?
intmat V[1][1] = 0; // all gen(i) of degree 0!
M = setModuleGrading(M, V);
hilb(std(M));
==> // 1 t^0
==> // -1 t^5
==>
==> // 1 t^0
==> // 1 t^1
==> // 1 t^2
==> // 1 t^3
==> // 1 t^4
==> // dimension (affine) = 0
==> // degree (affine) = 5
def h = hilbertSeries(M); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -t_(1)^5+1
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
module N = x^5*gen(3);
kill V;
intmat V[1][3] = 0; // all gen(i) of degree 0!
N = setModuleGrading(N, V);
hilb(std(N));
==> // 3 t^0
==> // -1 t^5
==>
==> // 3 t^0
==> // -1 t^5
==> // dimension (proj.) = 0
==> // degree (proj.) = 2
def h = hilbertSeries(N); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -t_(1)^5+3
==> -t_(1)+1
kill h;
////////////////////////////////////////////////
setring R;
module S = M + N;
S = setModuleGrading(S, V);
hilb(std(S));
==> // 3 t^0
==> // -2 t^5
==>
==> // 3 t^0
==> // -2 t^5
==> // dimension (proj.) = 0
==> // degree (proj.) = 1
def h = hilbertSeries(S); setring h;
==> ------------
==> This proc returns a ring with polynomials called 'numerator1/2' and 'deno\
minator1/2'!
==> They represent the first and the second Hilbert Series.
==> The s_(i)-variables are defined to be the inverse of the t_(i)-variables.
==> ------------
numerator1; denominator1;
==> -2*t_(1)^5+3
==> -t_(1)+1
kill h;
kill V;
kill R, W;
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