Complexity of Linear Equations and Infinite Gadgets

TL;DR

This work investigates the descriptive set-theoretic complexity of solving Borel systems of linear equations over a finite field. It proves that the solvability problem is uniformly $\mathbf{\Sigma}^1_2$-complete by a two-stage reduction: first establishing the hardness of infinite hypergraph colorings, then encoding these colorings into linear systems via infinite gadgets inspired by the Kechris-Solecki-Todorčević graph. The results hold first for $\mathbb{F}_2$ and extend to any finite field of characteristic $p$, illustrating a sharp divergence from the classical CSP dichotomy in the Borel context. This suggests a fundamental role for infinite combinatorial gadgets and motivates developing a theory of Borel polymorphisms to illuminate the boundary between easy and hard Borel CSPs.

Abstract

We investigate the descriptive set-theoretic complexity of the solvability of a Borel family of linear equations over a finite field. Answering a question of Thornton, we show that this problem is already hard, namely $Σ^1_2$-complete. This implies that the split between easy and hard problems is at a different place in the Borel setting than in the case of the CSP Dichotomy.

Theorems & Definitions (42)

• Theorem 1.1
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Theorem 2.7
• Definition 3.1
• Theorem 3.2