Computational Complexity of Polynomial Subalgebras
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.
Keywords
Cite
@article{arxiv.2502.05278,
title = {Computational Complexity of Polynomial Subalgebras},
author = {Leonie Kayser},
journal= {arXiv preprint arXiv:2502.05278},
year = {2025}
}
Comments
17 pages, comments welcome! Improved exposition in section 1. Major revision of section 4, proving the (previously conjectural) EXPSPACE-completeness. Accepted in Proceedings of ISSAC'25