Maximal Generating Degrees of Powers of Homogeneous Ideals

Abstract

The degree excess function 𝜖(I; n) is the difference between the maximal generating degree d(In) of the n-th power of a homogeneous ideal I of a polynomial ring and p(I)n, where p(I) is the leading coefficient of the asymptotically linear function d(In). It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal I whose 𝜖(I; n) has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on 𝜖(I; n) is provided. As an application, it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal I must be at least an exponential function of the number of variables.