Sign conditions for injectivity of generalized polynomial maps with\n applications to chemical reaction networks and real algebraic geometry

2013/11/21 by Stefan C. Müller, Elisenda Feliu, Müller, Stefan +9 · 3 citations
Chemistry · Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Axial and Atropisomeric Chirality Synthesis #Dynamical Systems (math.DS) #FOS: Mathematics #Molecular spectroscopy and chirality #Polynomial and algebraic computation #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1311.5493

openalex publication_date 2013/11/21 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We give necessary and sufficient conditions in terms of sign vectors for the\ninjectivity of families of polynomial maps with arbitrary real exponents\ndefined on the positive orthant. Our work relates and extends existing\ninjectivity conditions expressed in terms of Jacobian matrices and\ndeterminants. In the context of chemical reaction networks with power-law\nkinetics, our results can be used to preclude as well as to guarantee multiple\npositive steady states. In the context of real algebraic geometry,our work\nrecognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with\nwork of two of the authors, as the first partial multivariate generalization of\nthe classical Descartes' rule, which bounds the number of positive real roots\nof a univariate real polynomial in terms of the number of sign variations of\nits coefficients.\n

Citations

• Preclusion of switch behavior in reaction networks with mass-action\n kinetics
• General mass action kinetics
• Generalized Mass-Action Systems and Positive Solutions of Polynomial\n Equations with Real and Symbolic Exponents
• Some geometrical aspects of control points for toric patches

Cited by

• An effective criterion for multiple positive zeros of vertically parametrized polynomial systems
• Analog computation with transcriptional networks
• Degeneracy of Two-Dimensional Zero-One Reaction Networks with Up to Three Species

Related

• Geometry of the signed support of a multivariate polynomial and Descartes' rule of signs
• On the bijectivity of families of exponential/generalized polynomial\n maps
• Descartes' Rule of Signs for Polynomial Systems supported on Circuits
• On generalizing Descartes' rule of signs to hypersurfaces
• Lower bounds for positive roots and regions of multistationarity in\n chemical reaction networks