Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry
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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.
KeywordsSign vector Restricted injectivity Power-law kinetics Descartes’ rule of signs Oriented matroid
Mathematics Subject Classification13P15 12D10 70K42 37C10 80A30 52C40
This project began during the Dagstuhl Seminar on “Symbolic methods for chemical reaction networks” held in November 2012 at Schloss Dagstuhl, Germany. The authors also benefited from discussions during the AIM workshop on “Mathematical problems arising from biochemical reaction networks” held in March 2013, in Palo Alto. EF was supported by a postdoctoral grant “Beatriu de Pinós” from the Generalitat de Catalunya and the Spanish research project MTM2012-38122-C03-01. CC was supported by BMBF grant Virtual Liver (FKZ 0315744) and the research focus dynamical systems of the state Saxony-Anhalt. AS was supported by the NSF (DMS-1004380 and DMS-1312473). AD was partially supported by UBACYT 20020130100207BA, CONICET PIP 11220110100580, and ANPCyT PICT-2013-1110, Argentina. The authors thank an anonymous referee for helpful comments.
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