A Szemer\'edi-Trotter type theorem, sum-product estimates in finite quasifields, and related results

TL;DR

This paper extends Szemerédi-Trotter type theorems and sum-product estimates to finite quasifields, broadening the scope of combinatorial and algebraic results previously known for finite fields.

Contribution

It generalizes key combinatorial and algebraic results from finite fields to finite quasifields, including sum-product estimates and solvability of specific equations.

Findings

Established a Szemerédi-Trotter type theorem for quasifields

Derived sum-product estimates in finite quasifields

Generalized solvability results for equations over finite fields

Abstract

We prove a Szemer\'edi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and S\'ark\"ozy on the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields.