An algebraic approach to Latin squares of prime power order by local permutation polynomials

TL;DR

The paper addresses the problem of describing Latin squares of prime-power order by local permutation polynomials (LPPs) in $\,\mathbb{F}_q[x,y]$ and shows how LPP coefficients correspond to zeros of algebraic sets defined by zero-dimensional radical ideals. It provides a constructive method to represent all Latin squares of order $q$ with a single polynomial in $\,\mathbb{F}_q[x,y]$, supported by explicit computations for $q\in\{4,5\}$ and translations to reduced and isotopic classes, as well as to complete mappings and orthomorphisms. The work develops an algebraic-geometry framework using ideals $\mathcal{I}_q$ and $\mathcal{J}_q$ to describe coefficient constraints, studies isotopisms and isolinearity to classify LPPs, and extends the discussion to transversals and Hadamard products, including concrete $q=4$ analyses. The results enable systematic counting and classification of Latin squares via LPPs, with potential applications in combinatorial design, cryptography, and quasigroup theory. Overall, the paper provides a novel, constructive bridge between algebraic geometry and Latin-square theory, offering tools to enumerate, classify, and relate Latin squares through polynomial descriptions.

Abstract

Every Latin square of prime power order $q$ is uniquely described by a local permutation polynomial (LPP) in the polynomial ring $\mathbb{F}_q[x,y]$. Despite this equivalence, one may find in the literature only some preliminary results on the relationship among Latin squares and LPPs. This paper delves into this topic by showing how the coefficients of any LPP are identified with the zeros of an algebraic set over $\mathbb{F}_q$. This allows for an algebraic description of all Latin squares of order $q$ by means of a unique polynomial in $\mathbb{F}_q[x,y]$, whose coefficients satisfy the constraints defined by the algebraic set under consideration. In order to make much easier the construction of this polynomial, we also deal with the natural translation to LPPs of both notions of reduced and isotopic Latin squares. Our algebraic approach is readily adapted to identify both types of Latin squares. All of the above is constructively illustrated for $q\in\{4,5\}$. We finish our study with the natural translation to LPPs of both notions of complete mappings and orthomorphisms of quasigroups, showing their relationship with transversals and isotopisms of Latin squares.

Theorems & Definitions (46)

• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Remark 3.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.1