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The valuative capacity of the set of sums of \(d\)-th powers. (English) Zbl 1419.11050

Summary: If \(E\) is a subset of the integers then the \(n\)-th characteristic ideal of \(E\) is the fractional ideal of \(\mathbb{Z}\) consisting of 0 and the leading coefficients of the polynomials in \(\mathbb{Q}[x]\) of degree no more than \(n\) which are integer valued on \(E\). For \(p\) a prime the characteristic sequence of \(\text{Int}(E,\mathbb{Z})\) is the sequence \(\alpha_E(n)\) of negatives of the \(p\)-adic valuations of these ideals. The asymptotic limit \(\lim_{n\to\infty} \frac{\alpha_{E,p}(n)}{n}\) 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 \(\ell \geq 2\) integers to the power of \(d\), by observing the \(p\)-adic closure of these sets.

MSC:

11C08 Polynomials in number theory
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:

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