Khovanskii bases for semimixed systems of polynomial equations – Approximating stationary nonlinear Newtonian dynamics
Abstract
We provide an approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools rely on the theory of Khovanskii bases, combined with toric geometry, the Bernstein–Khovanskii–Kushnirenko (BKK) Theorem, and fiber products. As a direct application of this theory, we solve the problem of counting the number of approximate stationary states for coupled driven nonlinear resonators. We set up a system of polynomial equations that depends on three numbers N,n and M and whose solutions model the stationary states. The parameter N is the number of coupled resonators, 2n−1 is the degree of nonlinearity of the underlying differential equation, and M is the number of frequencies used in the approximation. We use our main theorems, that is, the generalized BKK Theorem 2.5 and the Decoupling Theorem 3.8, to count the number of (complex) solutions of the polynomial system for an arbitrary degree of nonlinearity 2n−1⩾3, any number of resonators N⩾1, and M=1 harmonic. We also solve the case N=1,n=2 and M=2 and give a computational way to check the number of solutions for N=1,n=2 and M>2. This extends the results of [1]. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000649790Publication status
publishedExternal links
Journal / series
Journal de Mathématiques Pures et AppliquéesVolume
Pages / Article No.
Publisher
ElsevierSubject
Khovanskii bases; BKK theorem; Fiber products; Multigraded Hilbert functionsMore
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