A Characterization of the Number of Roots of Linearized and Projective Polynomials in the Field of Coefficients

TL;DR

This paper provides a recursive characterization of the number of roots of linearized and projective polynomials over finite fields, linking root counts to matrix ranks smaller than Dickson matrices.

Contribution

It introduces a main theorem that characterizes root counts via the rank of a smaller matrix, advancing understanding of polynomial roots over finite fields.

Findings

Root count characterized by matrix rank

Recursive formulas for root enumeration

Connection to Dickson matrix properties

Abstract

A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive nature. These results follow from our main theorem which characterizes the number of roots using the rank of a matrix that is smaller than the Dickson matrix.