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Abstract
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The degree excess function e(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 e(I ; n) has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on e(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.
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