Existence of a unique, nondegenerate solution to parametrized systems of generalized polynomial equations

Abstract

We consider parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $x\in {\mathbb{R}}_{>}^n$ such that $A(c\circ x^B)=0$ with coefficient matrix $A\in {\mathbb{R}}^{l\times m}$, exponent matrix $B\in {\mathbb{R}}^{n\times m}$, parameter vector $c\in {\mathbb{R}}_{>}^m$ (and componentwise product $\circ$). Our main result characterizes the existence of a unique, nondegenerate solution (up to an exponential manifold) for all parameters in terms of the relevant geometric objects of the polynomial system: the $\text{coefficient polytope}$ and the $\text{monomial dependency subspace}$. Technically, we show that unique existence of a nondegenerate solution is equivalent to a composite (monomial-exponential moment) map being a diffeomorphism, and we characterize this property using Hadamard's global inversion theorem. Additionally, we provide sufficient conditions in terms of sign vectors of the geometric objects, which represent a genuine multivariate generalization of Descartes' rule of signs for exactly one solution. Finally, we illustrate all objects and results in a concrete example.