Computational Complexity of Polynomial Subalgebras

TL;DR

This paper analyzes the computational complexity of polynomial subalgebras, establishing EXPSPACE-completeness for the subalgebra membership problem and PSPACE-completeness for homogeneous algebras, highlighting key differences from ideal theory.

Contribution

It provides the first complexity classifications for subalgebra problems, including EXPSPACE-completeness and PSPACE-completeness results, advancing understanding in computational algebra.

Findings

Subalgebra membership problem is EXPSPACE-complete.

Homogeneous algebra membership problem is PSPACE-complete.

Differences between subalgebras and ideals are elucidated.

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 show completeness for 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.