Bivariate local permutation polynomials, their companions, and related enumeration results

TL;DR

This paper introduces a new family of bivariate local permutation polynomials over finite fields, constructs their companions, and provides exact enumeration results for these and related polynomial classes, advancing understanding of permutation polynomials.

Contribution

It constructs a new family of permutation group polynomials, explicitly finds their companions, and solves the enumeration problem for e-Klenian polynomials over finite fields.

Findings

Explicit construction of a new family of permutation polynomials.

Exact enumeration formulas for these polynomials.

Resolution of the enumeration problem for e-Klenian polynomials.

Abstract

We construct a new family of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For this family, we explicitly construct their companion. We also determine the total number of permutation group polynomials of this form. Moreover, we resolve the problem of enumerating $e$-Klenian polynomials over finite fields for $e\geq 1$, a problem previously noted as nontrivial by Gutierrez and Urroz (2023). In addition, we provide the exact number of permutation group polynomials equivalent to our proposed permutation group polynomials, as well as the exact number of those permutation group polynomials equivalent to $e$-Klenian polynomials.