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

2025-03-22

AI Summary

This work addresses the intrinsic computational complexity of solving systems of algebraic equations. We introduce “Galois width” — a novel complexity measure quantifying solution difficulty within finite computational models restricted to field operations and polynomial root adjunctions. Methodologically, we integrate Galois theory, algebraic geometry, and monodromy group analysis to derive computable lower bounds on complexity and expose fundamental limitations of classical heuristics—such as monodromy continuation. Our principal contribution is the first Galois-structural complexity criterion for multisolvable algebraic problems arising in geometric modeling, optimization, statistical inference, and computer vision. This criterion enables principled algorithm selection: identifying low-width instances amenable to accelerated solving, or rigorously proving the inevitable failure of standard numerical or symbolic methods.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Measure complexity of solving algebraic equations
• Link Galois width to practical heuristics like monodromy
• Apply Galois width to geometry, optimization, statistics, vision

Innovation

• Introduces Galois width for complexity measurement
• Uses field arithmetic and polynomial roots
• Links Galois width to practical heuristics