A characterization of the number of roots of linearized and projective polynomials in the field of coefficients

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.

A characterization of linearized polynomials with maximum kernel

“…During the "Combinatorics 2018" conference, the fourth author presented the results of this paper in the talk entitled "On q-polynomials with maximum kernel". In the same conference John Sheekey presented a joint work with Gary McGuire [8] in his talk entitled "Ranks of Linearized Polynomials and Roots of Projective Polynomials". It turned out that, independently from the authors of the present paper, they also obtained similar results.…”

“…It is important to have explicit conditions on the coefficients of a linearized polynomial characterizing the number of its roots. Further connections with projective polynomials can be found in [8].…”

Identifiers for MRD-codes

“…It can be seen that the rank of the matrix D f equals the rank of the F q -linear map ϕ, see for example [38] and also [9,17,27]. By the above remarks, it is straightforward to see that any F q -linear RMcode might be regarded as a suitable F q -subspace of L n,q .…”

On decoding additive generalized twisted Gabidulin codes

“…Furthermore, the necessary and sufficient condition for L(x) with q-degree k to have q k roots in F q n was independently characterized recently in [28,Theorem 7] and [7,Theorem 1.2], where all coefficients of L(x) are involved.…”