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We study the generic dimension of the zero set over $\mathbb{C}^*$,$\mathbb{R}^*$, and $\mathbb{R}_{>0}$ of parametric polynomial systemsconsisting of linear combinations of monomials scaled by free parameters. Thesesystems generalize sparse systems with fixed monomial support and freelyvarying parametric coefficients. As our main result, we establish theequivalence among three key properties: the existence of nondegenerate zeros,the zero set having the expected dimension, and the system having zeros forparameters in a Zariski dense subset of parameter space. Systems of this formdescribe the steady states of reaction networks modeled with mass-actionkinetics, and we use our results to answer several fundamental geometricquestions on topics such as generic finiteness, the difference between kineticand stoichiometric subspaces, absolute concentration robustness, andnondegenerate multistationarity.

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