Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

Abstract

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

MAPK’s networks and their capacity for multistationarity due to toric steady states

“…In particular, they are very useful for addressing matters of steady state characterization [17,24]. The same analysis we performed in the present work could be applied to many other important biochemical networks as long as they present toric steady states.…”

“…Even for mass-action systems, the large number of interacting species and the lack of knowledge of the reaction rate constants http become major drawbacks. If, however, the steady state ideal of the system is a binomial ideal, it was shown in [27] -and recently generalized in [24] -that these questions can be answered easily. Such systems are said to have toric steady states.…”

Topological and Statistical Methods for Complex Data

“…Polynomial equations are ubiquitous in numerous applications, such as algebraic statistics [29], chemical reaction kinetics [42], discretization of partial differential equations [28], satellite orbit design [47], circuit complexity [36], and cryptography [10]. The need to solve larger and larger equations, in applications as well as for theoretical purposes, has helped shape algebraic geometry and numerical analysis for centuries.…”

“…The triangle is a set of three vertices that are all connected to each other. There is much research on getting scalable algorithms for this problem; some focus on enumeration [6,9,12,32,38] and others on fast counting schemes [2,26,33,[42][43][44][45]. Diverse fields such as physics, sociology, biology, cybersecurity, computer science have focused on triangle counting [3,10,13,15,17,20,24,48].…”

Injectivity conditions of rational Bézier surfaces

“…Floater [24] proved that convex combination mappings are one-to-one if the boundary of the triangulation is homeomorphic to a convex polygon. Recently, Müller et al [19] gave the conditions of injectivity of the polynomial mappings based on the theory of algebraic geometry and combinatorics.…”