Complexity of Linear Equations and Infinite Gadgets

TL;DR

This paper proves that determining the solvability of Borel families of linear equations over finite fields is a highly complex problem, classified as -complete, revealing a different complexity landscape than classical CSP problems.

Contribution

It establishes the -completeness of the problem, answering Thornton's question and highlighting a distinct complexity boundary in the Borel setting.

Findings

The problem is -complete in the descriptive set-theoretic hierarchy.

The complexity of solving Borel linear equations differs from the CSP Dichotomy.

The result advances understanding of complexity in descriptive set theory and logic.

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 $\Sigma^1_2$-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.