Maximal Generating Degrees of Powers of Homogeneous Ideals

The degree excess function 𝜖 ( I ; n ) is the difference between the maximal generating degree d ( I n ) 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 ( I n ). 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.