On generating sets of minimal length for polynomial ideals (Q1820208)

Let \(A=K[X_ 1,...,X_ n]\) be the polynomial ring in n indeterminates over a field K. For an ideal I of A, let \(\mu(I)\) denote the smallest integer such that I can be generated by \(\mu(I)\) elements. The authors find a class of counter-examples where a set of generators of I corresponding to a homogeneous set of generators of the associated homogeneous ideal (or even the so called Gröbner basis) does not contain a generating set of I of cardinality \(\mu(I).\) The authors also discuss the problem of finding an algorithm to determine \(\mu(P)\) for a prime ideal P of A.