Computing positive tropical varieties and lower bounds on the number of positive roots

TL;DR

This paper introduces tools for computing positive tropicalizations of algebraic varieties, providing methods to estimate positive real roots and analyze steady states in chemical networks.

Contribution

It develops conditions and algorithms for positive tropicalization and bounds on positive roots, extending tropical geometry to real algebraic contexts.

Findings

Conditions for using initial ideals in positive tropicalization

A real version of the Transverse Intersection Theorem

Algorithm for bounding positive roots in polynomial systems

Abstract

We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to the Fundamental Theorem of Tropical Geometry. Additionally, under certain technical assumptions, we provide a real version of the Transverse Intersection Theorem. Building on these results, we propose an algorithm to compute a combinatorial bound on the number of positive real roots of a parametrized polynomial equations system. Furthermore, we discuss how this combinatorial bound can be applied to study the number of positive steady states in chemical reaction networks.