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We provide an approach to counting roots of polynomial systems, where eachpolynomial 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 thenumber of approximate stationary states for coupled driven nonlinearresonators. We set up a system of polynomial equations that depends on threenumbers $N, n$ and $M$ and whose solutions model the stationary states. Theparameter $N$ is the number of coupled resonators, $2n - 1$ is the degree ofnonlinearity of the underlying differential equation, and $M$ is the number offrequencies used in the approximation. We use our main theorems, that is, thegeneralized BKK Theorem and the Decoupling Theorem, to count the number of(complex) solutions of the polynomial system for an arbitrary degree ofnonlinearity $2n - 1 \geq 3$, any number of resonators $N \geq 1$, and $M = 1$harmonic. We also solve the case $N = 1, n = 2$ and $M = 2$ and give acomputational way to check the number of solutions for $N = 1, n = 2$ and $M\geq 2$. This extends the results of arXiv:2208.08179.

Khovanskii bases for semimixed systems of polynomial equations -- a case of approximating stationary nonlinear Newtonian dynamics