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Factorization of quaternionic polynomials of bi-degree (n,1)
Johanna Lercher1, Daniel Scharler2, Hans-Peter Schröcker1
1Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Technikerstr. 13, Innsbruck, 6020 Austria.
Beitrage Zur Algebra Und Geometrie
|February 27, 2023
Summary
We explore unique non-unique factorizations of bivariate polynomials over quaternions. These factorizations are geometrically and algebraically characterized by special rulings on associated ruled surfaces.
Area of Science:
- Algebra
- Geometry
- Quaternion Analysis
Background:
- Polynomials over non-commutative rings like quaternions present unique factorization challenges.
- Existing conditions for factorization into linear factors by Skopenkov and Krasauskas are recalled.
Purpose of the Study:
- To investigate and characterize novel types of non-unique factorizations for bivariate polynomials over quaternions.
- To establish a geometric and algebraic understanding of these special factorizations.
Main Methods:
- Analysis of bivariate polynomials of bi-degree (n, 1) over the skew field of quaternions.
- Geometric interpretation using ruled surfaces and their rulings (left/right) in projective space.
- Algebraic characterization through commutation properties of factors.
Main Results:
- Existence of bivariate quaternionic polynomials with non-unique factorizations beyond known cases.
- Geometric and algebraic characterization of these polynomials.
- Connection between special non-uniqueness and the degeneration of left/right rulings on the parameterized surface.
Conclusions:
- The study reveals new insights into the factorization properties of quaternionic polynomials.
- Geometric properties of ruled surfaces provide a powerful tool for understanding algebraic structures.
- Commutation properties of factors are key to explaining special non-uniqueness in factorizations.
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