Effective Results on non-Archimedean Tropical Discriminants

TL;DR

This paper investigates non-Archimedean A-discriminants, providing algorithms for their computation, bounds for specific cases, and tools for visualization, advancing understanding of tropical discriminants in p-adic contexts.

Contribution

It introduces an algorithm for computing non-Archimedean A-discriminants, establishes bounds for the case m=2, and offers a Sage package for visualizing discriminant amoebae.

Findings

Algorithm for non-Archimedean A-discriminant computation

Quadratic bounds for m=2 case

Sage package for plotting p-adic discriminant amoebae

Abstract

We study A-discriminants from a non-Archimedean point of view, refining earlier work on the tropical discriminant. In particular, we study the case where $A$ is a collection of n+m+1 points in Z^n in general position, and give an algorithm to compute the image of the A-discriminant variety under the non-Archimedean evaluation map. When m=2, our approach yields tight lower and upper bounds, of order quadratic in n. We also detail a Sage package for plotting certain p-adic discriminant amoebae, and present explicit examples of point sets yielding discriminant amoebae with extremal behavior.