On Factoring Quantum-Plane Skew Polynomials over Q(ω)(t)

Q: What mathematical object is targeted for factorization in the document?

Factorization is studied for dilation skew polynomials in a quantum-plane algebra of the form R over a function field associated with a cyclotomic field, with K=Q(ω) and a finite-order automorphism σ.

Q: How does the method use the centre to simplify the factorization task?

The algorithm first computes a monic bound in the commutative centre, then factors central left multiples and recursively handles multiplicities of central factors, reducing the hard remaining case to an irreducible central-bound situation.

Q: What is the two-level modular approach and why is it used?

The approach performs characteristic-zero computations long enough to exploit centre and cyclic-algebra structure, then specializes the central parameter to good algebraic values to descend to cyclic algebras over number fields. It further reduces at good inert primes so finite-field skew-factorization algorithms apply, after which factors are lifted back.

Q: What changes when enlarging constants from K to Q?

Stronger decidability results arise for a different factorization problem: in an algebraically closed constant-field setting, the irreducible-bound algebra splits by Tsen’s theorem, leading to decidable full factorization in an exact algebraic model based on finite extensions and root adjunction.

Jul 16, 2026

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Algorithms for factoring dilation skew polynomials in the quantum-plane (dilation) setting are developed over a function field tied to a cyclotomic field. The algebra R=K(σ)[δ;σ] with K=Q(ω) is analyzed using central elements to factor central left multiples and obtain a partial decomposition in characteristic zero. A two-level modular strategy specializes the central parameter to good values, studies cyclic algebras over number fields, then reduces at inert primes to apply fast finite-field skew-factorization and lift factors back. SageMath prototypes are provided, and extending constants to Q enables decidable factorization in an exact algebraic model.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

Introduction

Algebraic setup and central structure

Algorithmic approach and modular strategy

Constant-field extensions and decidability

Related work and further sections

arXiv :2607 .0275 1v 1 [ cs . SC] 2 Jul 2026 On Factoring Quantum-Plane Skew Polynomials over Q (􀀸)(􀁴) Mark Giesbrecht Cheriton School of Computer Science, Faculty of Mathematics, University of Waterloo, Canada [mwg@uwaterloo.ca](mwg@uwaterloo.ca) Abstract. We study algorithms for factorization in the quantum plane of (dilation) skew polynomials over a function field of a cyclotomic field: R = K (􀁃)[􀁇; 􀁦], K = Q (􀁬), 􀁦 (􀁃) = 􀁬􀁃, where 􀁬 ∈ C is a primitive 􀀼-th root of unity. We start with the established approach through central elements and factor the central left multiples, staying in characteristic zero, to obtain a partial decomposition. A two-level modular approach is proposed: specialize a central parameter to good algebraic values, study the resulting cyclic algebras over number fields, and then reduce further at good inert primes so that fast finite-field skew-factorization algorithms apply. A prototype SageMath implementation is provided to experiment with the algorithms. We then look at the effect of extending the field of constants from Q (􀁬) to Q, an algebraic closure of Q, and factoring over Q (􀁃)[􀁇; 􀁦] . In this case we show factorization is decidable in the exact algebraic model based on finite extensions. 1 Introduction This paper studies factorization of skew polynomials over a function field over a cyclotomic field with a dilation automorphism. This is a characteristic-zero analogue of the Ore polynomial setting over F􀁀 (􀁃) that underlies earlier skew factorization algorithms. In the root-of-unity quantum-plane case, the large univariate centre makes a “bound first” approach effective. Let K = Q (􀁬), where 􀁬 ∈ C is a primitive 􀀼-th root of unity, and let R = K (􀁃)[􀁇; 􀁦], 􀁇􀀰 = 􀁦 (􀀰)􀁇 (for any 􀀰 ∈ K (􀁃)), 􀁦(􀁃) = 􀁬􀁃 . Because 􀁦 has finite order 􀀼, its fixed field is K (􀁃􀀼 ) and the centre of R is the commutative principal ideal domain C = K (􀁃􀀼 )[􀁇􀀼 ] = K(T )[X], T = 􀁃􀀼 , X = 􀁇􀀼 . The centre is a polynomial ring in the single central variable X over the rational function field K(T ) . To appear, Computer Algebra in Scientific Computation (CASC) conference, August 31– September 4, 2026, Bath, UK 2 Mark Giesbrecht For our algorithms the initial input should first be cleared of denominators, but the factorization stages are carried out for a monic associate. Let 􀀛in ∈ R \ {0} be the given input. Clearing denominators and removing scalar content chooses an integral representative 􀀛 ∈ K [􀁃][􀁇; 􀁦] and a unit 􀁗 ∈ K (􀁃)× with 􀀛 = 􀁗􀀛in . Put ℓ = lc 􀁇 (􀀛) and 􀀵 = 􀀛♯ := ℓ −1􀀛 . Then 􀀵 is monic and 􀀛in = 􀁗 −1ℓ 􀀵 . The scalar 􀁗−1ℓ is a unit of R, so 􀀛in and 􀀵 have the same right divisors and the same irreducibility status, up to multiplication by units. Below, unless explicitly stated otherwise, 􀀵 denotes this working monic associate. A factorization 􀀵 = 􀀵 1 􀀵 2 · · · 􀀵 􀀺 into irreducibles in R gives the corresponding factorization 􀀛in = 􀁗 −1ℓ 􀀵 1 􀀵 2 · · · 􀀵 􀀺 . Denominators in 􀁃 may necessarily occur in the factors even when the chosen integral representative 􀀛 has coefficients in K [􀁃] . We use right-divisibility conventions throughout: ℎ is a right divisor of 􀀶 if 􀀶 = 􀁀ℎ . We write gcrd (􀀰, 􀀱) for the monic greatest common right divisor, equivalently the monic generator of R􀀰 + R􀀱 . For ℎ ≠ 0, rrem (􀀶, ℎ) denotes the right remainder in the division 􀀶 = 􀁀ℎ + 􀁁, where 􀁁 = 0 or deg 􀁇􀁁 < deg 􀁇 ℎ . A nonzero 􀀵 ∈ R is bounded if R 􀀵 ∩ C ≠ 0, i.e., if 􀀵 has a nonzero central left multiple. This is automatic in R, since it is free over C and is a domain. Following Jacobson (1943), the bound of 􀀵 is the unique monic 􀁩 ∈ C ∩ R 􀀵 of minimal degree in X, say 􀁩 = 􀁄 􀀵 with 􀁄 ∈ R. Such a bound is unique and computable by linear algebra. The overall algorithmic approach initially follows that of Giesbrecht (1998) and subsequent algorithms. First we compute a monic bound 􀁩 ∈ C. If 􀁩 factors in C, then right gcd computations split 􀀵 into a rough factorization whose pieces have irreducible ce

FAQs

What mathematical object is targeted for factorization in the document?

Factorization is studied for dilation skew polynomials in a quantum-plane algebra of the form R over a function field associated with a cyclotomic field, with K=Q(ω) and a finite-order automorphism σ.

How does the method use the centre to simplify the factorization task?

The algorithm first computes a monic bound in the commutative centre, then factors central left multiples and recursively handles multiplicities of central factors, reducing the hard remaining case to an irreducible central-bound situation.

What is the two-level modular approach and why is it used?

The approach performs characteristic-zero computations long enough to exploit centre and cyclic-algebra structure, then specializes the central parameter to good algebraic values to descend to cyclic algebras over number fields. It further reduces at good inert primes so finite-field skew-factorization algorithms apply, after which factors are lifted back.

What changes when enlarging constants from K to Q?

Stronger decidability results arise for a different factorization problem: in an algebraically closed constant-field setting, the irreducible-bound algebra splits by Tsen’s theorem, leading to decidable full factorization in an exact algebraic model based on finite extensions and root adjunction.

Jul 16, 2026

Upload by Aurora