Bivariate polynomial mappings associated with simple complex Lie algebras

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.

arXiv comment16 pages, 6 figures