The following functions deal with homogeneous modules. Homogeneity depends on the column weights, just as in graded polynomial rings.

MinimalBasis(M) : ModMPol -> [ ModMPolElt ]

Given a homogeneous module M, return a minimal basis of M. A basis is minimal if no proper subset of it generates the same submodule as it.

MinimalBasis(S) : [ ModMPolElt ] -> [ ModMPolElt ]

MinimalBasis(S) : { ModMPolElt } -> [ ModMPolElt ]

Given a set or sequence S of homogeneous module elements from a module M, return a minimal basis of the submodule of M generated by S. A basis is minimal if no proper subset of it generates the same submodule as it.

Given a homogeneous submodule M of the generic module G, return the Hilbert series H_(G/M)(t) of the quotient module G/M as an element of the univariate function field over the ring of integers.

> P<x, y, z> := PolynomialRing(RationalField(), 3);
> M := Module(P, 3);
> S := sub<M | [x, 0, z], [x^5 + y^5, z^5, y^3*x^2]>; 
> IsHomogeneous(S);
true
> H<t> := HilbertSeries(S);
> H;
1/(t^2 - 2*t + 1)