2024/09/17 by Abhishek Deshpande, Stefan C. Müller, Deshpande, Abhishek +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2409.11288
openalex publication_date 2024/09/17 · openalex created_date 2024/10/24 · openalex updated_date 2026/07/28
We consider parametrized systems of generalized polynomial equations (with real exponents) in n positive variables, involving m monomials with positive parameters; that is, x∈ℝn_> such that A (c ∘ xB)=0 with coefficient matrix A∈ℝl × m, exponent matrix B∈ℝn × m, parameter vector c∈ℝm_> (and componentwise product ∘). 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 coefficient polytope and the 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.