Matroidal root structure of skew polynomials over finite fields

Abstract

A skew polynomial ring R = K[x; σ, δ] is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A polynomial in such a ring may have more roots than its degree, which leads to the concepts of closures and independent sets of roots. There is also a structure of conjugacy classes on the roots. In R = Fqm [x, σ], this leads to the matroids Mr and M l of right independent and left independent sets, respectively. The matroids Mr and M l are isomorphic via the extension of the map φ : [1] → [1] defined by φ(a) = a m , where i = q i−1 −1 q−1 is a notation introduced to simplify the exponents in evaluation polynomials. Additionally, extending the field of coefficients of R results in a new skew polynomial ring S of which R is a subring, and if the extension is taken to include roots of an evaluation polynomial of f (x) (which does not depend on which side roots are being considered on), then all roots of f (x) in S are in the same conjugacy class.