Value sets of bivariate folding polynomials over finite fields

TL;DR

This paper determines the number of distinct outputs (value sets) of specific polynomial maps related to Lie algebras over finite fields, using roots of unity to analyze fixed points.

Contribution

It provides the first explicit calculation of value set sizes for bivariate folding polynomials associated with Lie algebras B2 and G2 over finite fields.

Findings

Cardinality formulas for B2 and G2 polynomial maps

Characterization of fixed points via roots of unity

Enhanced understanding of polynomial value sets over finite fields

Abstract

We find the cardinality of the value sets of polynomial maps associated with simple complex Lie algebras $B_2$ and $G_2$ over finite fields. We achieve this by using a characterization of their fixed points in terms of sums of roots of unity.