Barton, Andrea (2023-04). An Extension of Descartes' Rule of Signs to Multiple Variables, with Applications to Models of Biological Systems. Master's Thesis.
Thesis
Many biological systems are modeled mathematically using systems of differential equations. Descartes' rule of signs is a common tool for analyzing these systems. In this thesis we explore a new perspective on Descartes' rule of signs which allows us to begin extending the rule to polynomials in multiple variables, as introduced in two papers by Mate Telek and Elisenda Feliu. We explain the relationship between Descartes' rule of signs and this new perspective in detail, with several examples. Then we present a result for the situation when a polynomial in multiple variables has only one negative coefficient, with proof. Finally, we present some additional thoughts and potential areas for future study.
