On the bijectivity of families of exponential/generalized polynomial maps

TL;DR

This paper characterizes when families of generalized polynomial/exponential maps are bijective, ensuring unique solutions for parameterized systems, with applications to chemical reaction networks, using linear algebraic criteria and sign vector analysis.

Contribution

It provides effective criteria for bijectivity of generalized polynomial/exponential maps and analyzes robustness of solution uniqueness under perturbations.

Findings

Bijectivity characterized by sign vectors of coefficient and exponent subspaces.

Conditions for robustness of unique solutions under small perturbations.

Application of criteria to chemical reaction networks with mass-action kinetics.

Abstract

We start from a parametrized system of $d$ generalized polynomial equations (with real exponents) for $d$ positive variables, involving $n$ generalized monomials with $n$ positive parameters. Existence and uniqueness of a solution for all parameters and for all right-hand sides is equivalent to the bijectivity of (every element of) a family of generalized polynomial/exponential maps. We characterize the bijectivity of the family of exponential maps in terms of two linear subspaces arising from the coefficient and exponent matrices, respectively. In particular, we obtain conditions in terms of sign vectors of the two subspaces and a nondegeneracy condition involving the exponent subspace itself. Thereby, all criteria can be checked effectively. Moreover, we characterize when the existence of a unique solution is robust with respect to small perturbations of the exponents or/and the coefficients. In particular, we obtain conditions in terms of sign vectors of the linear subspaces or, alternatively, in terms of maximal minors of the coefficient and exponent matrices. Finally, we present applications to chemical reaction networks with (generalized) mass-action kinetics.