A characterization of linearized polynomials with maximum kernel

Csajbók, Bence and Marino, G. and Polverino, O. and Zullo, F. (2019) A characterization of linearized polynomials with maximum kernel. FINITE FIELDS AND THEIR APPLICATIONS, 56. pp. 109-130. ISSN 1071-5797

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Abstract

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over Fqn which ensure that the number of distinct roots of f in Fqn equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree qn−2 over Fqn 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 q-polynomial. Analogous results are proved for qs-polynomials as well, where gcd⁡(s,n)=1.

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