On the product of elements with prescribed trace

TL;DR

This paper investigates the conditions under which elements in finite field extensions can be expressed as products of elements with prescribed trace values, providing solutions for various extension degrees and applications in finite geometry and related areas.

Contribution

It offers a comprehensive solution for expressing elements as products with prescribed traces in finite fields, especially for extensions of degree at least 5, and extends results to certain arbitrary fields and degrees.

Findings

Complete solution for finite fields with extension degree ≥ 5.

Results for fields with extension degrees 2, 3, and 4.

Applications to PN functions, semifields, and finite geometry.

Abstract

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\mathrm{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $x\cdot y$, where $x,y\in \mathbb{L}$ with $\mathrm{Tr}(x)=a, \mathrm{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least $5$. We also have results for arbitrary fields and extensions of degrees $2,3$ or $4$. We then apply our results to the study of PN functions, semifields, irreducible polynomials with prescribed coefficients, and to a problem from finite geometry concerning the existence of certain disjoint linear sets.