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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 leadingcoefficients 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$Int(E,\mathbb{Z})$ is the sequence $\alpha_E (n)$ of negatives of the $p$-adicvaluations of these ideals. The asymptotic limit $\lim_{n\to\infty}\frac{\alpha_{E,p}(n)}{n}$ of this sequence, called the valuativecapacity of $E$, gives information about the geometry of $E$. We compute thesevaluative capacities for the sets $E$ of sums of $\ell \geq 2$ integers to thepower of $d$, by observing the $p$-adic closure of these sets.

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