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

Abstract

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