On Factoring Quantum-Plane Skew Polynomials over $\mathbb{Q}(\ddot{I}‰)(t)$

Abstract

We study algorithms for factorization in the quantum plane of (dilation) skew polynomials over a function field of a cyclotomic field: \[ \mathsf{R}=\mathsf{K}(t)[x;σ], \qquad \mathsf{K}=\mathbb{Q}(ω), \qquad σ(t)=ωt, \] where $\ddot{I}‰\in \mathbb{C}$ is a primitive $m$-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 $\mathbb{Q}(\ddot{I}‰)$ to $\bar{\mathbb{Q}}$, an algebraic closure of $\mathbb{Q}$, and factoring over $\bar{\mathbb{Q}}(t)[x;\ddot{I}ƒ]$. In this case we show factorization is decidable in the exact algebraic model based on finite extensions.

To appear in Computer Algebra and Scientific Computation conference, August 31-September 4, Bath, UK