Khovanskii Bases for Semimixed Systems of Polynomial Equations

In this talk, I will present an efficient approach for counting roots of polynomial systems, where each polynomial is a general linear combination of fixed, prescribed polynomials. Our tools primarily rely on the theory of Khovanskii bases, combined with toric geometry.

I will demonstrate the application of this approach to the problem of counting the number of approximate stationary states for coupled Duffing oscillators. We have derived a Khovanskii basis for the corresponding polynomial system and determined the number of its complex solutions for an arbitrary degree of nonlinearity in the Duffing equation and an arbitrary number of oscillators. This is the joint work with Paul Breiding, Mateusz Michalek, Javier del Pino, and Oded Zilberberg.