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D.15.19.1 nchilb
Procedure from library ncHilb.lib (see ncHilb_lib).
- Usage:
- nchilb(list of relations, an integer, optional);
L is a list of modules (each module represents a free-polynomial),
d is an integer for the degree bound (maximal total degree of the
polynomials of generating set of input ideal),
#[]=1 represents the case of non-finitely generated ideals,
#[]=2 is for the computation of multi-graded Hilbert series, and
#[]=3 is to print the details about the orbit and system of equation.
- Note :
- The generating set of input ideal should be a Groebner basis and needs
to be given in a special form. It is a list of modules, where each
generator of every module represents a monomial times a coefficient
in the free associative algebra. The first entry, in each generator,
represents a coefficient and every next entry is a variable.
Ex: module p1=[1,y,z],[-1,z,y] represents the poly y*z-z*y;
module p2=[1,x,z,x],[-1,z,x,z] represents the poly x*z*x-z*x*z
for more details about the input, see examples.
Example:
| LIB "ncHilb.lib";
ring r=0,(X,Y,Z),dp;
module p1 =[1,Y,Z]; //represents the poly Y*Z
module p2 =[1,Y,Z,X]; //represents the poly Y*Z*X
module p3 =[1,Y,Z,Z,X,Z];
module p4 =[1,Y,Z,Z,Z,X,Z];
module p5 =[1,Y,Z,Z,Z,Z,X,Z];
module p6 =[1,Y,Z,Z,Z,Z,Z,X,Z];
module p7 =[1,Y,Z,Z,Z,Z,Z,Z,X,Z];
module p8 =[1,Y,Z,Z,Z,Z,Z,Z,Z,X,Z];
list l1=list(p1,p2,p3,p4,p5,p6,p7,p8);
nchilb(l1,10);
==>
==> Maximal length of words = 2
==>
==> Orbit length = 3
==>
==>
==> Hilbert series:
==> 1/(t2-3t+1)
==>
ring r=0,(x,y,z),dp;
==> // ** redefining r (ring r=0,(x,y,z),dp;)
module p1=[1,y,z],[-1,z,y]; //y*z-z*y
module p2=[1,x,z,x],[-1,z,x,z]; // x*z*x-z*x*z
module p3=[1,x,z,z,x,z],[-1,z,x,z,z,x]; // x*z^2*x*z-z*x*z^2*x
module p4=[1,x,z,z,z,x,z];[-1,z,x,z,z,x,x]; // x*z^3*x*z-z*x*z^2*x^2
==> x*gen(7)+x*gen(6)+x*gen(3)+z*gen(5)+z*gen(4)+z*gen(2)-gen(1)
list l2=list(p1,p2,p3,p4);
nchilb(l2,6,1); //third argument '1' is for non-finitely generated case
==>
==> Maximal length of words = 3
==>
==> Orbit length = 6
==>
==>
==> Hilbert series:
==> 1/(t3+t2-3t+1)
==>
ring r=0,(a,b),dp;
==> // ** redefining r (ring r=0,(a,b),dp;)
module p1=[1,a,a,a];
module p2=[1,a,b,b];
module p3=[1,a,a,b];
list l3=list(p1,p2,p3);
nchilb(l3,5,2);//third argument '2' is to compute multi-graded HS
==>
==> Maximal length of words = 3
==>
==> Orbit length = 5
==>
==>
==> Hilbert series:
==> (t(1)^2+t(1)+1)/(t(1)*t(2)^2-t(1)*t(2)-t(2)+1)
==>
ring r=0,(x,y,z),dp;
==> // ** redefining r (ring r=0,(x,y,z),dp;)
module p1=[1,x,z,y,z,x,z];
module p2=[1,x,z,x];
module p3=[1,x,z,y,z,z,x,z];
module p4=[1,y,z];
module p5=[1,x,z,z,x,z];
list l4=list(p1,p2,p3,p4,p5);
nchilb(l4,7,3); //third argument '3' is to print the details
==>
==> Maximal length of words = 3
==>
==> Orbit length = 6
==>
==> Words description of the Orbit:
==> 1 x(1) y(1) x(1)*z(2) y(1)*z(2) x(1)*z(2)*z(3)
==>
==> linear system:
==> H(1) = (t)*H(2) + (t)*H(3) + (t)*H(1) + 1
==> H(2) = (t)*H(2) + (t)*H(3) + (t)*H(4) + 1
==> H(3) = (t)*H(2) + (t)*H(3) + (t)*H(5) + 1
==> H(4) = (t)*H(5) + (t)*H(3) + (t)*H(6) + 1
==> H(5) = (t)*H(5) + (t)*H(5) + (t)*H(5) + 0
==> H(6) = (t)*H(3) + (t)*H(3) + (t)*H(1) + 1
==> where H(1) represents the series corresp. to input ideal
==> and i^th summand in the rhs of an eqn. is according
==> to the right colon map corresp. to the i^th variable
==>
==> Hilbert series:
==> (t3+t2+1)/(2t5-2t4-t3+2t2-3t+1)
==>
// of the orbit and system
ring r=0,(x,y,z),dp;
==> // ** redefining r (ring r=0,(x,y,z),dp;)
module p1=[1,y,z,z];
module p2=[1,y,y,z];
module p3=[1,x,z,z];
module p4=[1,x,z,y];
module p5=[1,x,y,z];
module p6=[1,x,y,y];
module p7=[1,x,x,z];
module p8=[1,x,x,y];
module p9=[1,y,z,y,z];
module p10=[1,y,z,x,z];
module p11=[1,y,z,x,y];
module p12=[1,x,z,x,z];
module p13=[1,x,z,x,y];
module p14=[1,x,y,x,z];
module p15=[1,x,y,x,y];
module p16=[1,y,z,y,x,z];
module p17=[1,y,z,y,x,y];
module p18=[1,y,z,y,y,x,z];
module p19=[1,y,z,y,y,x,y];
module p20=[1,y,z,y,y,y,x,z];
module p21=[1,y,z,y,y,y,x,y];
list l5=list(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,
p14,p15,p16,p17,p18,p19,p20,p21);
nchilb(l5,7,1,2,3);
==>
==> Maximal length of words = 3
==>
==> Orbit length = 7
==>
==> Words description of the Orbit:
==> 1 x(1) y(1) x(1)*x(2) y(1)*y(2) y(1)*z(2) x(1)*x(2)*y(3\
)
==>
==> linear system:
==> H(1) = (t(1))*H(2) + (t(2))*H(3) + (t(3))*H(1) + 1
==> H(2) = (t(1))*H(4) + (t(2))*H(4) + (t(3))*H(4) + 1
==> H(3) = (t(1))*H(2) + (t(2))*H(5) + (t(3))*H(6) + 1
==> H(4) = (t(1))*H(4) + (t(2))*H(7) + (t(3))*H(7) + 1
==> H(5) = (t(1))*H(2) + (t(2))*H(5) + (t(3))*H(7) + 1
==> H(6) = (t(1))*H(4) + (t(2))*H(6) + (t(3))*H(7) + 1
==> H(7) = (t(1))*H(7) + (t(2))*H(7) + (t(3))*H(7) + 0
==> where H(1) represents the series corresp. to input ideal
==>
==> Hilbert series:
==> (-t(1)*t(2)-t(1)*t(3)-t(2)*t(3)-1)/(t(1)*t(2)*t(3)-t(1)*t(2)-t(1)*t(3)+t(\
1)-t(2)*t(3)+t(2)+t(3)-1)
==>
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