Z/pZ中的一个加法定理与极大零和自由集
An addition theorem and maximal zero-sum free sets in Z/pZ
摘要：
利用加性数论中的多项式方法，建立了$\mathbb{Z}/p\mathbb{Z}$中满足$A\cap(-A)=\emptyset$的集合的子和集的一个新的加法定理：\[|Σ(A)|\geqslant\min{p,1+\frac{|A|(|A|+1)}{2}}.\]该证明在性质上类似于Alon，Nathanson和Ruzsa对Erdös-Heilbronn猜想的证明（最初由Dias da Silva和Hamidoune \cite{DH}证明）.在证明这一定理的一个关键点是评价一些二项式行列式已研究的工作中的Gessel和Viennot。一个推广到一个序列的子和集的派生，导致零和自由序列的结构结果。作为另一个应用，证明了对于任意素数$p$，$\mathbb{Z}/p\mathbb{Z}$中的最大零和自由集具有最大整数$k$的基数，使得\[\frac{k(k+1)}{2}<p,\]证明了Selfridge的一个猜想.
英文摘要：
Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying $A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb{Z}$: \[|Σ(A)|\geqslant\min{p,1+\frac{|A|(|A|+1)}{2}}.\] The proof is similar in nature to Alon, Nathanson and Ruzsa's proof of the Erdös-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune \cite{DH}). A key point in the proof of this theorem is the evaluation of some binomial determinants that have been studied in the work of Gessel and Viennot. A generalization to the set of subsums of a sequence is derived, leading to a structural result on zero-sum free sequences. As another application, it is established that for any prime number $p$, a maximal zero-sum free set in $\mathbb{Z}/p\mathbb{Z}$ has cardinality the greatest integer $k$ such that \[\frac{k(k+1)}{2}<p,\] proving a conjecture of Selfridge from 1976.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
