Generic root counts of tropically transverse systems -- An invitation to tropical geometry in OSCAR

Abstract

The main mathematical focus of this paper is a class of parametrised polynomial systems that we refer to as being tropically transverse. We show how their generic number of solutions can be expressed as the mixed volume of a modified system. We then provide an alternate proof of a recent result by Borovik et al, on the number of equilibria of coupled nonlinear oscillators using elementary tropical geometry. The proof draws upon a wide range of concepts across tropical geometry, which we will use as an opportunity to give a first overview over various tropical features in OSCAR.