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

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• 作者：Marie-André ; e B. Langlois ^{mblanglois@dal.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Integer valued polynomials ; Valuative capacity ; p-Orderings ; Sums of powers
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：155-168
• 全文大小：346 K

文摘

If E is a subset of the integers then the n-th characteristic ideal of E   is the fractional ideal of Z$\mathbb{Z}$ consisting of 0 and the leading coefficients of the polynomials in Q[x]$\mathbb{Q}[x]$ of degree no more than n which are integer valued on E. For p   a prime the characteristic sequence of Int(E,Z)$\mathit{Int}(E,\mathbb{Z})$ is the sequence α_E(n)${\alpha }_E(n)$ 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 ℓ≥2$\ell \ge 2$ integers to the power of d, by observing the p-adic closure of these sets.