The quadratic Poisson Gel’fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polynomial algebra K[X1, . . . , Xn] with Poisson bracket defined by {Xi, Xj} = λijXiXj for some skew-symmetric matrix (λij) ∈ Mn(K). This problem was studied in [9] over a field of characteristic 0 by using a Poisson ve...

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring Tn(R,α). By using an ideal theory of a skew triangular matrix ring Tn(R,α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x;α]/〈xn〉, for each positive integer n, where R[x;α] is the skew polynomial ring, and 〈xn〉 is the ideal generated by xn.

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers xn for binary encoded numbers n. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to coRNC, which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented...

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann-Tzeng bound that works for a wide ...

This paper is concerned with ideals in a commutative Noetherian ring R of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of R generated by regular sequences exhibit a desirable type of ‘uniform’ behaviour. The principal technical tool used is a result, proved by R. Hartshorne and R. Speiser in the case where R is local and contains its residue fi...

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of...

In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the...

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were i...

We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified defini...