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The Prouhet--Tarry--Escott problem for subsets with small doubling in integral domains

  • Ernie Croot
  • Junzhe Mao
  • Chi Hoi Yip
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Abstract

The Prouhet--Tarry--Escott (PTE) problem has many generalizations and has been studied in various algebraic domains. In this paper, we prove that finite subsets S of integral domains with small additive doubling constant (but still a power of |S|) always contain solutions to Wright's generalization of the PTE problem: there are small subsets A and B of the same size such that ∑a∈ A aj=∑b∈ B bj for 1≤ j≤ k, but not for j=k+1. More generally, our method gives simultaneous solutions for m systems, with pairwise distinct (k+1)-th power sums. In contrast with the classical case S⊆ [N], where the problem has been studied by Wooley and others using Vinogradov's mean value theorem, our approach is based on polynomial identities and additive properties of S. We also discuss barriers to extending these results to broader settings.

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Cite

BibTeXRIS
BibTeX
@misc{croot2026prouhettarryescott,
  title = {{The Prouhet--Tarry--Escott problem for subsets with small doubling in integral domains}},
  author = {Ernie Croot and Junzhe Mao and Chi Hoi Yip},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05061},
  eprint = {2609.05061},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05061},
}
RIS
TY  - UNPB
TI  - The Prouhet--Tarry--Escott problem for subsets with small doubling in integral domains
AU  - Ernie Croot
AU  - Junzhe Mao
AU  - Chi Hoi Yip
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05061
UR  - https://arxiv.org/abs/2609.05061
AB  - The Prouhet--Tarry--Escott (PTE) problem has many generalizations and has been studied in various algebraic domains. In this paper, we prove that finite subsets S of integral domains with small additive doubling constant (but still a power of |S|) always contain solutions to Wright's generalization of the PTE problem: there are small subsets A and B of the same size such that ∑_a∈ A a^j=∑_b∈ B b^j for 1≤ j≤ k, but not for j=k+1. More generally, our method gives simultaneous solutions for m systems, with pairwise distinct (k+1)-th power sums. In contrast with the classical case S⊆ [N], where the problem has been studied by Wooley and others using Vinogradov's mean value theorem, our approach is based on polynomial identities and additive properties of S. We also discuss barriers to extending these results to broader settings.
ER  -