Factorizations for difference operators
TL;DR: In this article, the authors consider the factorization of regular polynomials in the ring of difference operators and obtain an analogue of the fundamental theorem of algebra for skew polynomial ring over field K.
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Abstract: We consider the factorization problems of difference operators in $\mathbb{C}[x;\sigma]$
for an automorphism σ of finite order. We study the factorization of regular polynomials in $\mathbb{R}[x]$
in the ring of such difference operators and obtain an analogue of the fundamental theorem of algebra for skew polynomial ring $\mathsf{K}[x; \sigma]$
over field K.
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Citations
•Journal Article
Petit algebras and their automorphisms
TL;DR: In this paper, the authors study the properties of a nonassociative algebra construction from skew polynomial rings, and show that it is solvable if and only if it can be written as a chain of Petit algebras satisfying certain conditions.
16
How to obtain lattices from (f,sigma,delta)-codes via a generalization of Construction A
TL;DR: In this paper, it was shown that cyclic (f,σ,δ)-codes over finite rings can induce a Z-lattice in RN by using certain quotients of orders in non-associative division algebras defined using the skew polynomial f. This construction generalizes the one using certain σ-constacyclic codes by Ducoat and Oggier.
Nontrivial solutions of second-order singular Dirichlet systems
Jin Zhao,Yanchao Wang +1 more
TL;DR: In this article, the existence of nontrivial solutions for second-order singular Dirichlet systems is proved based on the Leray-Schauder nonlinear alternative principle, which is based on a well-known fixed point theorem in cones.
Addendum to “Factoring skew polynomials over Hamilton's quaternion algebra and the complex numbers” [J. Algebra 427 (2015) 20–29]
TL;DR: In this paper, the quaternion division algebra over a real closed field F is shown to decompose every non-constant polynomial in a skew-polynomial ring D [ t ; σ, δ ] into a product of linear factors, and thus has a zero in D.
1
Darboux transformations and Fay identities for the extended bigraded Toda hierarchy
Bojko Bakalov,Anila Yadavalli +1 more
TL;DR: In this article, a bilinear equation for the tau-function of the extended bigraded Toda hierarchy (EBTH) was derived and generalized Fay identities were derived from it.
References
•Book
Noncommutative Noetherian Rings
J. McConnell,J. Robson +1 more
- 27 Feb 2001
TL;DR: The history of mathematics can be surveyed from many dierent perspectives, such as those that try to shed light on the history of particular theorems and on the people who created them as mentioned in this paper.
2.3K
•Book
Finite-dimensional division algebras over fields
Nathan Jacobson
- 21 Oct 1996
TL;DR: Skew Polynomials and Division Algebras are used in the Brauer Factor Sets and Noether Factor Sets as mentioned in this paper, as well as Galois Descent and Generic Splitting Fields.
343
Vandermonde and Wronskian matrices over division rings
Tsit Yuen Lam,André Leroy +1 more
TL;DR: The basic facts on skew polynomials in the (S, D)-setting are developed, the evaluation of such polynomial ring K is defined, and the all-important “Product Theorem” is proved.
167