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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
##### Software:
QSopt_ex; RAGlib; SageMath; QSopt-Exact; TOPCOM
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##### References:
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