Speaker
Description
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 $B \in \mathbb R^{n \times m}$, and parameter vector $c \in \mathbb R^m_>$ (and with componentwise product $\circ$).
We demonstrate that the existence of a unique nondegenerate solution for all parameters is equivalent to a specific moment map being a diffeomorphism. We characterize this property using Hadamard's global inversion theorem and establish sufficient conditions based on the sign vectors of the underlying geometric objects. This result represents a multivariate generalization of Descartes' rule of signs for exactly one nondegenerate solution.
