发表机构
Georgia Institute of Technology; Hong Kong University of Science and Technology（佐治亚理工学院; 香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究整环中具小加法加倍常数的有限子集S，证明其包含Wright推广的PTE问题的解，方法基于多项式恒等式与S的加法性质，还探讨了结果推广的障碍。
AI 中文摘要
Prouhet–Tarry–Escott（PTE）问题有诸多推广形式，已在各类代数整环中得到研究。本文证明：整环的有限子集S若具有小加法加倍常数（且仍为|S|的幂），则必包含Wright推广的PTE问题的解：存在大小相同的小子集A和B，使得对1≤j≤k有∑_{a∈A}a^j=∑_{b∈B}b^j，而对j=k+1不成立。更一般地，本文方法可给出m个方程组的同时解，其(k+1)次幂和两两不同。与经典情形S⊆[N]（Wooley等人用Vinogradov均值定理研究该情形）不同，本文方法基于多项式恒等式和S的加法性质，还讨论了将这些结果推广到更广泛场景的障碍。
英文摘要
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 $\sum_{a\in A} a^j=\sum_{b\in B} b^j$ for $1\le j\le k$, 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 $S\subseteq [N]$, 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.
Comments19 pages
