Efficient algorithms are presented for factoring polynomials in the skew-polynomial ring F[x; σ], a non-commutative generalization of the usual ring of polynomials F[x], where F is a finite field and σ: F → F is an automorphism (iterated Frobenius map). Applications include fast functional decomposition algorithms for a class of polynomials in F[x] whose decompositions are “wild” and previously...

The work of Dixmier in 1977 and Moeglin in 1980 show us that for a prime ideal P in the universal enveloping algebra of a complex finite-dimensional Lie algebra the properties of being primitive, rational and locally closed in the Zariski topology are all equivalent. This equivalence is referred to as the Dixmier-Moeglin equivalence. In this thesis we will study skew Laurent polynomial rings of...

A skew Laurent polynomial ring S = R[x;α] is reversible if it has a reversing automorphism, that is, an automorphism θ of period 2 that transposes x and x and restricts to an automorphism γ of R with γ = γ. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent...

let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...

where 7-[ is the category of finitely presented E-torsion R-modules of projective dimension 1. An explicit construction for 0 is given, generalizing that previously worked out by Bass when ~ is central. Application is made to work of Farrell and Hsiang on skew polynomial tings, and to the Whitehead groups of those division rings that occur as fields of fractions ofnoncommutative Euclidean domains.

No general algorithm is known for the functional decomposition of wild polynomials over a finite field. However partial solutions exist. In particular, a fast functional decomposition algorithm for linearised polynomials has been developed using factoring methods in skew-polynomial rings. This algorithm is extended to a related class of wild polynomials, which are sub-linearised polynomials.