Bivariate polynomial mappings associated with simple complex Lie algebras

TL;DR

This paper investigates conditions under which certain bivariate polynomial maps linked to rank-2 simple complex Lie algebras induce permutations over finite fields, extending known results and providing counterexamples to existing conjectures.

Contribution

It extends permutation criteria to polynomial maps associated with B2 and G2 Lie algebras and addresses a conjecture related to Schur's problem.

Findings

Permutation criteria for B2 and G2 polynomial maps established

Counterexample provided to Lidl and Wells' conjecture

Conditions depend on gcd relations with finite field sizes

Abstract

There are three families of bivariate polynomial maps associated with the rank-$2$ simple complex Lie algebras $A_2, B_2 \cong C_2$ and $G_2$. It is known that the bivariate polynomial map associated with $A_2$ induces a permutation of $\mathbf{F}_q^2$ if and only if $\gcd(k,q^s-1)=1$ for $s=1, 2, 3$. In this paper, we give similar criteria for the other two families. As an application, a counterexample is given to a conjecture posed by Lidl and Wells about the generalized Schur's problem.