The valuative capacity of the set of sums of d-th powers

Abstract

If E is a subset of the integers then the n-th characteristic ideal of E is the fractional ideal of Z consisting of 0 and the leading coefficients of the polynomials in Q[x] of degree no more than n which are integer valued on E. For p a prime the characteristic sequence of Int(E,Z) is the sequence αE(n) of negatives of the p-adic valuations of these ideals. The asymptotic limit limn→∞⁡αE,p(n)n of this sequence, called the valuative capacity of E, gives information about the geometry of E. We compute these valuative capacities for the sets E of sums of ℓ≥2 integers to the power of d, by observing the p-adic closure of these sets.