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

TL;DR

This paper investigates the valuative capacity of sets formed by sums of $d$-th powers of integers, linking algebraic properties with geometric insights through $p$-adic analysis.

Contribution

It introduces a method to compute the valuative capacity of sum-of-$d$-th powers sets using $p$-adic closure analysis, connecting algebraic and geometric perspectives.

Findings

Computed valuative capacities for sums of $ ext{ell} extgreater= 2$ $d$-th powers.

Established a link between $p$-adic closure and the geometric structure of these sets.

Provided explicit formulas or methods for capacity calculation.

Abstract

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 $Int(E,\mathbb{Z})$ 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.