We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time group algebra setting rank-metric codes and in Hamming metric. allows to construct twisted linearized Reed-Solomon new family maximum distance extending Sheekey's Gabidulin rank Furthermore, we analogue Trombetti-Zhou construction, which also pro...

This paper introduces methods of pseudo-linear algebra to unify the algebraic formalism of one-forms and the related polynomial approach for both continuous and discrete-time nonlinear control systems. Given approach covers also differene, q-shift and q-difference operators whereby this algebraic formalism is not only unified but also extended to wide class of nonlinear control systems. Also th...

Abstract. Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded...

Let D be a division ring, T be a variable over D, σ be an endomorphism of D, δ be a σ-derivation on D and R = D[T ;σ, δ] the left skew polynomial ring over D. We show that the set (V alν(R), ) of σ-compatible real valuations which extend to R a fixed proper real valuation ν on D has a natural structure of parameterized complete non-metric tree, where is the partial order given by μ μ̃ if and onl...

We apply new techniques to compute Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets on polyvector fields. We obtain as consequences some conditions under which brackets are always 0, and show that the Hochs...

In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise description of quotients of skew polynomial rings by a left principal ideal, using results relating skew polynomial rings to Azumaya algebras. We use this des...

We describe Poincaré–Birkhoff–Witt bases for the two-parameter quantum groups U = Ur,s(sln) following Kharchenko and show that the positive part of U has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on U , which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to ...

We consider separably closed fields of characteristic p > 0 and fixed Imperfection degree as modules over a skew polynomial ring. We axlotnatlze the correspondltlg theory and ue show that i t is complete and that ~t adnilts quantifier e l ~ n i ~ n a t ~ o n In the usual nlodule language augmented with additwe functions whlch are the analog of the p-component functions. $