Value sets of bivariate Chebyshev maps over finite fields

TL;DR

This paper calculates the number of distinct outputs (value sets) of bivariate Chebyshev maps over finite fields by analyzing their dynamical behavior and fixed points expressed via roots of unity.

Contribution

It provides a new method to determine the value set cardinality of bivariate Chebyshev maps using dynamical systems and algebraic fixed point analysis.

Findings

Cardinality of value sets over finite fields determined

Fixed points characterized via roots of unity

Method applicable to other polynomial maps

Abstract

We determine the cardinality of the value sets of bivariate Chebyshev maps over finite fields. We achieve this using the dynamical properties of these maps and the algebraic expressions of their fixed points in terms of roots of unity.