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

Sartaj Ul Hasan, Ramandeep Kaur, Hridesh Kumar

TL;DR

The paper develops two new families of bivariate local permutation polynomials over $\

Abstract

We introduce two new families of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For each family, we explicitly construct their companions. Furthermore, we precisely determine the total number of permutation group polynomials equivalent to the proposed families. Moreover, we resolve the problem of enumerating permutation group polynomials that are equivalent to $e$-Klenian polynomials over finite fields for $e\geq 1$, a problem previously noted as nontrivial by Gutierrez and Urroz (2023).

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

TL;DR

The paper develops two new families of bivariate local permutation polynomials over $\

Abstract

We introduce two new families of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For each family, we explicitly construct their companions. Furthermore, we precisely determine the total number of permutation group polynomials equivalent to the proposed families. Moreover, we resolve the problem of enumerating permutation group polynomials that are equivalent to -Klenian polynomials over finite fields for , a problem previously noted as nontrivial by Gutierrez and Urroz (2023).

Paper Structure

This paper contains 5 sections, 19 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutation group polynomials
  4. The companions
  5. Enumeration of equivalent permutation group polynomials

Key Result

Lemma 2.1

JJ_2023 There is a bijective map between the set of local permutation polynomials $f\in \mathbb F_q[X,Y]$ and the set of permutation polynomial tuples $\underline{\beta}_f :=(\beta_0,\ldots,\beta_{q-1}) \in {\mathfrak S}_q^q$. Furthermore, $f$ and $\underline{\beta}_f$ are associated to each other b

Theorems & Definitions (40)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 30 more