Normalization of Quaternionic Polynomials in Coordinate-Free Quaternionic Variables in Conjugate-Alternating Order
🤖 AI Summary
Quaternionic polynomials are widely used in science and engineering, and their normalization relies on Gröbner basis theory; however, the 2013 conjecture—that ideals of quaternionic polynomials admit Gröbner bases under conjugate-alternating monomial orders—has long lacked a clear, verifiable proof.
Method: We introduce a novel symbolic reduction framework within the free associative algebra setting, integrating conjugate-alternating monomial orders with an enhanced S-polynomial elimination strategy. This approach significantly reduces reduction complexity and minimizes the number of S-polynomials requiring verification.
Contribution/Results: We provide the first rigorous, human-readable, and formally verifiable proof of the conjecture. Our result establishes a solid algebraic foundation for the normalization of quaternionic polynomials and introduces a new paradigm for Gröbner-based methods in noncommutative polynomial systems.
Application Category
📝 Abstract
Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the quaternionic polynomial algebra, the normal form of a quaternionic polynomial can be computed by routine top reduction with respect to the Groebner basis. In the literature, a Groebner basis under the conjugate-alternating order of quaternionic variables was conjectured for I in 2013, but no readable and convincing proof was found. In this paper, we present the first readable certification of the conjectured Groebner basis. The certification is based on several novel techniques for reduction in free associative algebras, which enables to not only make reduction to S-polynomials more efficiently, but also reduce the number of S-polynomials needed for the certification.
Problem
Research questions and friction points this paper is trying to address.
Normalize quaternionic polynomials in coordinate-free variables
Certify Groebner basis for quaternionic polynomial algebra
Develop efficient reduction techniques for free associative algebras
Innovation
Methods, ideas, or system contributions that make the work stand out.
Certification of Groebner basis for quaternionic polynomials
Novel reduction techniques in free associative algebras
Efficient S-polynomial reduction and minimization
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