Matroidal Root Structure of Skew Polynomials over Finite Fields

Abstract

A skew polynomial ring $R=K[x;\sigma,\delta]$ 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 $R=F_{q^m}[x,\sigma]$, this leads to matroids of right independent and left independent sets. These matroids are isomorphic via the extension of the map $\phi:[1]\to[1]$ defined by $\phi(a)=a^{\frac{q^{i-1}-1}{q-1}}$. 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.