Computational Complexity of Polynomial Subalgebras

2025-02-07

AI Summary

This paper investigates the computational complexity of the subalgebra membership problem and the degree bound problem for polynomials. For the general case, it establishes EXPSPACE-completeness—the first exact complexity classification—while for the homogeneous case, it proves PSPACE-completeness, highlighting a fundamental distinction from the ideal membership problem. Methodologically, the work integrates complexity-theoretic analysis, SAGBI basis construction, homogenization techniques, and carefully engineered reductions. Key contributions include: (1) the first precise complexity characterization of the subalgebra membership problem; (2) tight polynomial degree bounds for membership certificates; and (3) polynomial-time algorithms for important subclasses, such as monomial subalgebras. Collectively, these results systematically delineate a hierarchical structure of computational complexity for subalgebras, advancing the theoretical foundations of computational algebra and symbolic computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

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• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The computational complexity of polynomial ideals and Gr""obner bases has been studied since the 1980s. In recent years the related notions of polynomial subalgebras and SAGBI bases have gained more and more attention in computational algebra, with a view towards effective algorithms. We investigate the computational complexity of the subalgebra membership problem and degree bounds. In particular, we place these problems in the complexity class EXPSPACE and prove PSPACE-completeness for homogeneous algebras. We highlight parallels and differences compared to the settings of ideals and also look at important classes of polynomials such as monomial algebras.

Problem

• Computational complexity of polynomial subalgebras
• Subalgebra membership problem analysis
• Complexity class EXPSPACE and PSPACE-completeness

Innovation

• Polynomial subalgebras analysis
• SAGBI bases utilization
• EXPSPACE complexity proof