A characterization of linearized polynomials with maximum kernel

Abstract

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over F q n which ensure that the number of distinct roots of f in F q n equals the degree of f . We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree q n−2 over F q n which have maximum kernel and for n ≤ 6 we list all q-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary qpolynomial. Analogous results are proved for q s -polynomials as well, where gcd(s, n) = 1.

Scalar q-subresultants and Dickson matrices

“…Results in a similar direction have been obtained recently in [5] where for each q-polynomial f of q-degree k, k conditions were given, in terms of the coefficients of f , which are satisfied if and only if f has rank n´k (there is a hidden pk`1q-th condition here as well, namely the assumption that the coefficient of x q k in f is non-zero). Independently, in [16] it was proved that the rank of f is n´m if and only if a certain kˆk matrix has rank k´m.…”

New results on quasi-subfield polynomials

“…The following result (independently due to McGuire and Sheekey [3] and Csajbók et al [4]) characterizes linearized polynomials that split completely.…”

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.…”

“…In [28] the authors also applied companion matrices to study the number of roots of the above linearized polynomial. Further works on the roots of linearized polynomials can be found in [7,32,46].…”