SubalgebraBases in Macaulay2
Michael Burr, Oliver Clarke, Timothy Duff, Jackson Leaman, Nathan Nichols, Elise Walker
TL;DR
The paper revisits the SubalgebraBases package in Macaulay2, detailing its support for computing subalgebra bases (SAGBI, canonical, and Khovanskii bases) for polynomial rings and quotients. It introduces two key data structures, Subring and SAGBIBasis, to model subalgebras and to track/resume partial computations via the sagbi computation object, with a rich set of options (e.g., $Limit$, engine vs Top subduction, and strategy choices). The work also describes the subduction framework, binomial syzygies, normal forms, and auxiliary tools such as $subringIntersection$ and $groebnerMembershipTest$, enabling robust manipulation and membership testing. Through diverse examples—including invariant theory, toric degenerations, and Grassmannians like $\mathrm{Gr}(3,6)$—the paper demonstrates practical applicability, extensibility, and the importance of resummable computations for large-scale algebraic geometry problems.
Abstract
We describe a recently revived version of the software package SubalgberaBases, which is distributed in the Macaulay2 computer algebra system. The package allows the user to compute and manipulate subagebra bases -- which are also known as SAGBI bases or canonical bases and form a special class of Khovanskii bases -- for polynomial rings and their quotients. We provide an overview of the design and functionality of SubalgberaBases and demonstrate how the package works on several motivating examples.