实代数集上$s$ -距离集大小的一个上界
An upper bound for the size of $s$-distance sets in real algebraic sets
摘要：
在最近的一份文件彼得罗夫和Pohoata开发了一种新的代数方法，结合了Croot-列夫-Pach引理从添加剂组合和西尔维斯特定律的惯性真正的二次型。作为应用，他们给出了$s$ -距离集（子集$\mbox{$ \cal A $}\subseteq {\mathbb R}^n$最多确定$s$不同距离）大小的Bannai-Bannai-Stanton界的简单证明。在本文中，我们推广了他们的工作，并证明了上界的大小$s$ -距离集在各种实代数集。这种方法得到了Delsarte-Goethals-Seidel在球面$s$ -距离集上的界的一个新的简短证明，并推广了Bannai-Kawasaki-Nitamizu-Sato在球面并上的$s$ -距离集上的界.在我们的论点中，我们使用彼得罗夫和Pohoata的方法与一些Gröbner基础技术。
英文摘要：
In a recent paper Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester's Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mbox{$\cal A$}\subseteq {\mathbb R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical $s$-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.
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