SAGBI detection and SAGBI homotopy

We present a SAGBI detection algorithm which, given a finite set of polynomials, determines all term orders for which this set forms a SAGBI basis. Based on this, we implement the SAGBI homotopy package in Julia using HomotopyContinuation.jl and Oscar.jl. The package provides a specific choice of start system to proceed with homotopy. For systems where each equation is a linear combination of fixed polynomials, SAGBI homotopies can significantly reduce the number of solution paths that need to be tracked compared to polyhedral homotopies. We will illustrate this approach on polynomial systems arising in applications from the sciences.

This is based on joint works with Elima Shehu, Timothy Duff, and Barbara Betti.