On the rational relationships among pseudo-roots of anon-commutative polynomial

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article

Retakh, Vladimir¹  Saks, Michael¹ 

[1] Rutgers State Univ, Dept Math, Piscataway, NJ 08854 USA

关键词: Non-commutative polynomials;    Factorizations;    Roots;    Lattices;   

DOI  :  10.1016/j.jpaa.2020.106581

来源: Elsevier

【 摘 要 】

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t) = p(0) + p(1)t +. . . p(k)t(k) is an element xi epsilon R, for which there exist polynomials q(1), q(2) such that p = q(1)(t-xi)q(2). We investigate the rational relationships that hold among the pseudo-roots of p(t) by using the diamond operations for cover graphs of modular lattices. When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial f(S) that vanishes on S. By results of Leroy and Lam [16], the set of polynomials {f(T) : T subset of S} with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set Lambda(S) of pseudo-roots of f(S). Given an arbitrary subset of Lambda(S), our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of Lambda(S), and in particular, rationally generates S. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free