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07 February 2025
 
  » arxiv » 1609.0299

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The valuative capacity of the set of sums of $d$-th powers
Marie-Andree B.Langlois ;
Date 1 Sep 2016
AbstractIf $E$ 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 $lim_{n o infty}frac{alpha_{E,p}(n)}{n}$ 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.
Source arXiv, 1609.0299
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