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

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.