论文标题
实际代数集的$ s $ distance集的上限
An upper bound for the size of $s$-distance sets in real algebraic sets
论文作者
论文摘要
在最近的一篇论文中，彼得罗夫（Petrov）和波霍塔（Pohoata）开发了一种新的代数方法，该方法结合了添加剂组合学和西尔维斯特（Sylvester）的惯性法的croot-lev-pach引理，以实现实际二次形式。作为一个应用程序，他们简单地证明了Bannai-Bannai-Stanton在$ s $ distance sets的大小（subsets $ \ mbox {$ \ cal a $} \ subseteq {\ mathbb r}^n $确定最多$ s $不同距离的大小）。在本文中，我们扩展了他们的工作，并证明了各种真实代数集的$ s $ distance集的上限。这样，我们为球形$ s $ distance套装的delsarte-coethals-seidel的界限获得了一个小说和简短的证明，并在$ s $ s $ distance in-s $ distance set sets of Spheres ban bannai-kawasaki-nitamizu-sato限制的概括。在我们的论点中，我们将Petrov和Pohoata的方法与一些Gröbner基础技术一起使用。
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.
