On Efficient Noncommutative Polynomial Factorization via Higman Linearization

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F < x 1 , x 2 , … , x n > of polynomials in noncommuting variables x 1 , x 2 , … , x n over the field F . We obtain the following result: Given a noncommutative algebraic branching program...

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Veröffentlicht in:Computational complexity Jg. 34; H. 2; S. 10
Hauptverfasser: Arvind, V., Joglekar, Pushkar S.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.12.2025
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Computation
Computational Mathematics and Numerical Analysis
Computer Science
Factorization
Fields (mathematics)
Linearization
Mathematical analysis
Polynomials
Rings (mathematics)
identity testing
Higman linearization
68 Computer Science
invariant subspaces
factorization
Noncommutative polynomials
linear matrices
ISSN:1016-3328, 1420-8954
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Zusammenfassung:In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F < x 1 , x 2 , … , x n > of polynomials in noncommuting variables x 1 , x 2 , … , x n over the field F . We obtain the following result: Given a noncommutative algebraic branching program of size s computing a noncommutative polynomial f ∈ F < x 1 , x 2 , … , x n > as input, where F = F q is a finite field, we give a randomized algorithm that runs in time polynomial in s , n and log 2 q that computes a factorization of f as a product f = f 1 f 2 ⋯ f r , where each f i is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming f into a linear matrix L using Higman linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f . We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing a nontrivial common invariant subspace of a collection of matrices over finite fields.
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content type line 14
ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-025-00272-9