A note on hyperquadratic elements of low algebraic degree

TL;DR

This paper investigates hyperquadratic elements of low algebraic degree over fields of finite characteristic, using differential algebra to analyze the structure of associated projective polynomials.

Contribution

It introduces a differential algebra method to study hyperquadratic elements and characterizes the factorization of projective polynomials over finite fields.

Findings

Projective polynomials over Fp have only linear or quadratic factors.

Hyperquadratic elements are linked to roots of specific projective polynomials.

The method restricts the form of polynomials associated with hyperquadratic elements.

Abstract

In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply linked to the characteristic p of K. Roots of these projective polynomials are particular algebraic elements over K, called hyperquadratic. For a general algebraic element of degree d over K, we discuss the possibility of being hyperquadratic. Using a method of differential algebra, we obtain, for particular fields K = Fp, projective polynomials only having polynomial factors of degree 1 or 2.