Abstract
AbstractWe consider polynomials of bi-degree (n, 1) over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a necessary and sufficient condition for existence of a factorization with univariate linear factors that has originally been stated by Skopenkov and Krasauskas. Such a factorization is, in general, non-unique by known factorization results for univariate quaternionic polynomials. We unveil existence of bivariate polynomials with non-unique factorizations that cannot be explained in this way and characterize them geometrically and algebraically. Existence of factorizations is related to the existence of special rulings of two different types (left/right) on the ruled surface parameterized by the bivariate polynomial in the projective space over the quaternions. Special non-uniqueness in above sense can be explained algebraically by commutation properties of factors in suitable factorizations. A necessary geometric condition for this to happen is degeneration to a point of at least one of the left/right rulings.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Algebra and Number Theory
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献