[Paper Review] Factorization of Non-Commutative Polynomials

Fabrizio Caruso|arXiv (Cornell University)|Feb 16, 2010

graph theory and CDMA systems2 references3 citations

TL;DR

This paper presents an algorithm for factoring non-commutative polynomials over a field, extending an unpublished sketch by James H. Davenport. It uses graded decomposition and symbolic extensions to handle non-unique factorizations in the inhomogeneous case, with a complete implementation in the Axiom computer algebra system and improvements that reduce computational complexity via commutative image analysis and overlap minimization.

ABSTRACT

We describe an algorithm for the factorization of non-commutative polynomials over a field. The first sketch of this algorithm appeared in an unpublished manuscript (literally hand written notes) by James H. Davenport more than 20 years ago. This version of the algorithm contains some improvements with respect to the original sketch. An improved version of the algorithm has been fully implemented in the Axiom computer algebra system.

Motivation & Objective

• To develop a practical and efficient algorithm for factoring non-commutative polynomials over a field, addressing the lack of published solutions despite prior theoretical sketches.
• To resolve the issue of non-uniqueness in factorizations of inhomogeneous polynomials by introducing symbolic extensions to model cancellations.
• To implement the algorithm in the Axiom computer algebra system, leveraging its support for free monoids and non-commutative polynomial rings.
• To reduce the exponential search space through two key improvements: minimizing algebraic extensions and using commutative images to filter impossible factorizations.
• To lay the groundwork for formal proof of finiteness of factorizations and exploration of alternative algebra quotients.

Proposed method

• The algorithm decomposes a non-commutative polynomial into homogeneous components and recursively determines factor components by matching monomials and solving for coefficients under cancellation constraints.
• It introduces new symbols (algebraic extensions) to represent possible cancellations between overlapping monomials in the product $GH$.
• The method uses left and right quotients to reconstruct candidate factors $G$ and $H$ from monomials in the original polynomial $F$.
• For each monomial $M$ in $F$, it checks whether a candidate head factor $\hat{G}$ left-divides $M$ or $\hat{H}$ right-divides $M$, updating $G$ and $H$ accordingly.
• The algorithm applies a recursive system of equations derived from homogeneous components $F_{n-j} = G_{h-j}H_k + \dots + G_h H_{k-j}$ to determine consistent factor structures.
• It uses the commutative image of the polynomial to pre-filter possible factor degree pairs, reducing the number of cases to explore via the binary knapsack problem.

Experimental results

Research questions

• RQ1 Can a systematic and efficient algorithm be designed for factoring non-commutative polynomials over a field, despite the non-uniqueness of factorizations?
• RQ2 How can cancellations between overlapping monomials in non-commutative products be modeled algebraically to enable factor reconstruction?
• RQ3 To what extent can the commutative image of a non-commutative polynomial be used to prune the search space for valid factorizations?
• RQ4 Is it possible to reduce the number of required algebraic extensions (symbols) in the coefficient field by strategic selection of initial monomial factors?
• RQ5 What conditions ensure the finiteness of non-commutative factorizations, and can this be formally proven?

Key findings

• The algorithm correctly factors non-commutative polynomials by decomposing them into homogeneous components and solving for factor coefficients through monomial matching and symbolic extensions.
• For the polynomial $F = yxyxy - y$, the algorithm finds two distinct factorizations: $(yx \pm 1)(yxy \mp y)$, derived from the condition $\alpha^2 = 1$.
• The use of commutative images reduces the number of candidate factor degree pairs by filtering via the binary knapsack problem on irreducible commutative factors.
• The algorithm’s implementation in Axiom supports symbolic computation of all possible factorizations and outputs them with explicit descriptions of coefficient constraints.
• The number of required algebraic extensions can be minimized by choosing initial monomial factors $\hat{G}$ and $\hat{H}$ with minimal overlap, reducing computational overhead.
• The algorithm is fully implemented and will be included in the next release of the Axiom computer algebra system.