A Jacobian criterion for the simultaneous injectivity on positive variables of linearly parameterized polynomial maps

Consider a map g : R-r x R-n -> R-P x R-m such that for k is an element of R-r and x is an element of R-n, g(k, x) = (L(x), f (k, x)), where L : R-n -> R-P is a linear map and f (k, x) = Sigma(r)(i=1) k(i) x(ai) v(i) = Sigma(r)(i-1) k(i) x(1)(a1i) ... x(n)(ain) v(i); a(i) is an element of Z(>= 0)(n) and V-i is an element of R-m are fixed for i = 1, ..., r. We prove that the partially evaluated mapg(k, -) : R-n -> R-p x R-m is injective on R->0(n) for every k is an element of R->0(r) if and only if for each k is an element of R->0(r) and x is an element of R->0(n) the (linear) derivative map D(g, k, x) : R-n -> R-p x R-m of g(k, -) at x is injective. This result is useful for studying the uniqueness or multiplicity of equilibria in conservative systems of chemical reactions under mass action. A map such as f would represent the rates of change of concentrations of all or some judiciously selected species. The linear map L would represent the time-invariant total concentrations. To illustrate this application, we prove the uniqueness of equilibria in a common pharmacological model of receptorligand interaction, without a customary assumption on rate constants that lets all equilibria be of a strong type known as detailed balance. Our result extends a theorem of Craciun and Feinberg applicable to maps of the kind off. That earlier result is directly applicable to models of chemical reactions that include the outflow of all species. (c) 2012 Elsevier Inc. All rights reserved.