Some results on linearized trinomials that split completely
Gary McGuire, Daniela Mueller · Finite Fields and Their Applications · 2020
Linearized polynomials over finite fields have been the subject of many papers over the last several decades. Recently, there has been a renewed interest in linearized polynomials because of new connections to coding theory and finite geometry. We consider the problem of calculating the rank or nullity of a linearized polynomial L(x) = Σdi =0 aixqi (where ai ∈ Fqn ) from the coefficients ai. The rank and nullity of L(x) are the rank and nullity of the associated Fq-linear map Fqn → Fqn . McGuire and Sheekey [6] defined a d × d matrix AL with the property that nullity(L) = nullity(AL − I). We present some consequences of this result for some trinomials that split completely, i. e., trinomials L(x) = xqd −bxq −ax that have nullity d. We give a full characterization of these trinomials for n ≤ d2 − d + 1.