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

TL;DR

This paper characterizes when parametrized systems of generalized polynomial equations have a unique positive solution, using geometric objects and a multivariate Descartes' rule of signs, advancing understanding of solution existence and uniqueness.

Contribution

It provides a geometric and algebraic characterization of the unique solution existence for parametrized generalized polynomial systems, including a multivariate Descartes' rule of signs.

Findings

Unique solution existence is characterized by the bijectivity of a moment map.

The geometric objects determine solution uniqueness and existence.

A multivariate Descartes' rule of signs is established for exactly one solution.

Abstract

We consider solutions to 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 $\circ$. As our main result, we characterize the existence of a unique solution (modulo an exponential manifold) for all parameters in terms of the relevant geometric objects of the polynomial system, namely the $\textit{coefficient polytope}$ and the $\textit{monomial dependency subspace}$. We show that unique existence is equivalent to the bijectivity of a certain moment/power map, and we characterize the bijectivity of this map using Hadamard's global inversion theorem. Furthermore, we provide sufficient conditions in terms of sign vectors of the geometric objects, thereby obtaining a multivariate Descartes' rule of signs for exactly one solution.