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

Consider a map g:Rr×Rn→Rp×Rmg:Rr×Rn→Rp×Rm such that for k∈Rrk∈Rr and x∈Rnx∈Rn, g(k,x)=(L(x),f(k,x))g(k,x)=(L(x),f(k,x)), where L:Rn→RpL:Rn→Rp is a linear map andf(k,x)=∑i=1rkixaivi=∑i=1rkix1ai1⋯xnainvi;ai∈Z⩾0n and vi∈Rmvi∈Rm are fixed for i=1,…,ri=1,…,r. We prove that the partially evaluated map g(k,-):Rn→Rp×Rmg(k,-):Rn→Rp×Rm is injective on R>0n for every k∈R>0r if and only if for each k∈R>0r and x∈R>0n, the (linear) derivative map D(g,k,x):Rn→Rp×RmD(g,k,x):Rn→Rp×Rm of g(k,-)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 receptor–ligand 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 of f. That earlier result is directly applicable to models of chemical reactions that include the outflow of all species.