# Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry Source: [Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry]() Collection index: [llms.txt]() Data updated: 2025-10-25T11:18:30.250Z Authors: Stefan M\\"uller, Elisenda Feliu, Georg Regensburger, Carsten Conradi, Anne Shiu, Alicia Dickenstein Published: 2013-11-21T17:43:57.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()