Geometry of Yang-Baxter matrix equations over finite fields

TL;DR

The paper studies the Yang–Baxter matrix equation for $2\times2$ matrices over the finite field $\mathbb{F}_q$, describing the affine solution variety $\mathcal{D}_A(\mathbb{F}_q)$ and its geometry via computational ideal theory. It reduces the problem to four rational canonical forms $A_1,A_2,A_3,A_4$ (with $A_4$ previously understood) and derives explicit solution sets and cardinalities for $\mathcal{D}_{A_1}(\mathbb{F}_q)$, $\mathcal{D}_{A_2}(\mathbb{F}_q)$, and $\mathcal{D}_{A_3}(\mathbb{F}_q)$ across parameter regimes, including discriminant-like quantities $\delta$ and $\Delta$. The results give precise descriptions of dimensions, irreducible components, and finite counts, and uncover a complete description in many cases while proposing conjectures for the remaining ones (notably $A_3$ with $\Delta\neq -b$). The work combines similarity/classification arguments, ideal-theoretic decompositions, and Gröbner-basis computations, supplemented by explicit small-field examples and MAGMA code.

Abstract

Let $A$ be a $2\times 2$ matrix over a finite field and consider the Yang-Baxter matrix equation $XAX=AXA$ with respect to $A$. We use a method of computational ideal theory to explore the geometric structure of the affine variety of all solutions to this equation. In particular, we exhibit all solutions explicitly and determine cardinality formulas for these varieties.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Corollary 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 2.5