Factorization of Quaternionic Polynomials of Bi-Degree (n,1)

TL;DR

This paper investigates the factorization properties of bi-degree (n,1) quaternionic polynomials, revealing conditions for existence, non-uniqueness, and geometric interpretations involving rulings on associated ruled surfaces.

Contribution

It provides a complete characterization of factorizability and non-uniqueness for quaternionic bi-degree (n,1) polynomials, linking algebraic and geometric conditions.

Findings

Existence of factorizations is characterized by geometric rulings.

Non-uniqueness of factorizations is explained by algebraic commutation properties.

Degeneration of rulings indicates potential non-uniqueness in factorizations.

Abstract

We 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.