Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. (English) Zbl 1382.92272
Summary: 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 G. Craciun et al. [Lect. Notes Comput. Sci. 5862, 111–135 (2010; Zbl 1274.65033)], 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.
MSC:
| 92E20 | Classical flows, reactions, etc. in chemistry |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 37C10 | Dynamics induced by flows and semiflows |
| 52C40 | Oriented matroids in discrete geometry |
| 80A30 | Chemical kinetics in thermodynamics and heat transfer |
Keywords:
sign vector; restricted injectivity; power-law kinetics; Descartes’ rule of signs; oriented matroidCitations:
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