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 $\textit{coefficient polytope}$ and the $\textit{monomial dependency subspace}$. 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.