A Galois-Theoretic Complexity Measure for Solving Systems of Algebraic Equations

TL;DR

This paper introduces the Galois width $gw(\alpha)$ as a Galois-theoretic, minimax measure of algebraic solving complexity within an algebraic-computation model that adjoins roots. It shows that $gw(\alpha)$ equals $gw(G)$ for the Galois group $G$ of the minimal polynomial, and that $gw(G)$ decomposes through composition factors as $gw(G)=\max_i\mu(N_i/N_{i+1})$, linking complexity to minimal faithful permutation degrees. The work connects this intrinsic measure to monodromy and thin-set results, providing lower bounds and guiding algorithmic strategies, including homotopy and decomposition-based approaches. Through detailed applications in metric algebraic geometry, algebraic statistics, and algebraic vision, the authors illustrate that $gw$ can be substantially smaller than traditional degree-based bounds, enabling more efficient solutions in structured problems and clarifying when radical solvability is possible or impossible.

Abstract

Motivated by applications of algebraic geometry, we introduce the Galois width, a quantity characterizing the complexity of solving algebraic equations in a restricted model of computation allowing only field arithmetic and adjoining polynomial roots. We explain why practical heuristics such as monodromy give (at least) lower bounds on this quantity, and discuss problems in geometry, optimization, statistics, and computer vision for which knowledge of the Galois width either leads to improvements over standard solution techniques or rules out this possibility entirely.

Figures (2)

• Figure 1: Minimizing the distance from a point in $\mathbb{R}^3$ to the surface of \ref{['ex:solid-rev']}.
• Figure 2: A 3D point (blue) and its 1-dimensional radial projection (red.)

Theorems & Definitions (54)

• Proposition 1
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• Lemma 1
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• Remark 1
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• Definition 1
• Proposition 2
• Theorem 1