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The main mathematical focus of this chapter 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 an overview of various tropical features in OSCAR.

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Generic Root Counts of Tropically Transverse Systems an Invitation to Tropical Geometry in OSCAR

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  • 发表日期：

    2025年

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  • Isaac Holt,
  • Yue Ren

摘要

The main mathematical focus of this chapter 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 an overview of various tropical features in OSCAR.