Dimension and degeneracy of zeros of parametric polynomial systems with applications to reaction networks

We study the generic dimension of the zero set over ℂ^*, ℝ^*, and ℝ_>0 of parametric polynomial systems consisting of linear combinations of monomials scaled by free parameters. These systems generalize sparse systems with fixed monomial support and freely varying parametric coefficients. As our main result, we establish the equivalence among three key properties: the existence of nondegenerate zeros, the zero set having the expected dimension, and the system having zeros for parameters in a Zariski dense subset of parameter space. Systems of this form describe the steady states of reaction networks modeled with mass-action kinetics, and we use our results to answer several fundamental geometric questions on topics such as generic finiteness, the difference between kinetic and stoichiometric subspaces, absolute concentration robustness, and nondegenerate multistationarity.