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We give necessary and sufficient conditions in terms of sign vectors for theinjectivity of families of polynomial maps with arbitrary real exponentsdefined on the positive orthant. Our work relates and extends existinginjectivity conditions expressed in terms of Jacobian matrices anddeterminants. In the context of chemical reaction networks with power-lawkinetics, our results can be used to preclude as well as to guarantee multiplepositive steady states. In the context of real algebraic geometry,our workrecognizes a prior result of Craciun, Garcia-Puente, and Sottile, together withwork of two of the authors, as the first partial multivariate generalization ofthe classical Descartes' rule, which bounds the number of positive real rootsof a univariate real polynomial in terms of the number of sign variations ofits coefficients.

Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry