SAGBI combinatorics of maximal minors and a SAGBI algorithm

The maximal minors of a matrix of indeterminates are a universal Grobner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal SAGBI basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to SAGBI bases. These experiments motivated a new implementation of the SAGBI algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2, version 1.21, and Singular, version 4.2.1 and a package intended for CoCoA 5.4.2, it extends the range of computability by at least one order of magnitude.& COPY; 2023 Elsevier Ltd. All rights reserved.

Automated summary

The maximal minors of a matrix of indeterminates are a universal Grobner basis according to a theorem by Bernstein, Sturmfels, and Zelevinsky. However, they are not always a universal SAGBI basis. Experimental findings on their behavior under varying monomial orders and their extension to SAGBI bases have motivated the development of a new implementation of the SAGBI algorithm using a Singular script and Normaliz for combinatorial computations. Compared to other packages, it significantly expands the range of computability.