Detecting tropical defects of polynomial equations

TL;DR

This paper introduces tropical defects as certificates for non-tropical basis systems and presents algorithms to find them, solving open problems in algebraic geometry related to del Pezzo surfaces and valuated gaussoids.

Contribution

It proposes the concept of tropical defects and provides algorithms to identify them, advancing the understanding of tropical bases and their applications.

Findings

Identified tropical defects for specific polynomial systems.

Solved open problems on del Pezzo surfaces of degree 3.

Addressed realizability issues of valuated gaussoids on 4 elements.

Abstract

We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.