Factorizations of certain bivariate polynomials

TL;DR

This paper completely characterizes the factorization of specific bivariate polynomials involving additive polynomials over fields of odd characteristic, addressing a question related to algebraic geometry and Kakeya problems.

Contribution

It provides the first complete factorization results for these polynomials over all fields of odd characteristic, solving an open problem posed by Slavov.

Findings

Explicit factorizations for X*f(X)-Y*g(Y) over K[X,Y]

Applicable to all squarefree additive polynomials in K[X]

Answers a key open question in algebraic geometry related to Kakeya problems

Abstract

We determine the factorization of X*f(X)-Y*g(Y) over K[X,Y] for all squarefree additive polynomials f,g in K[X] and all fields K of odd characteristic. This answers a question of Kaloyan Slavov, who needed these factorizations in connection with an algebraic-geometric analogue of the Kakeya problem.