  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.

Algebraic Geometry;Dynamical Systems