A point-conic incidence bound and applications over Fp. (January 2023)

• Record Type: Journal Article
• Authors: Mohammadi, Ali; Pham, Thang; Warren, Audie
• Is Part Of: European journal of combinatorics. Volume 107(2023)
• Publication Date: 2023-01
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2022.103596
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.