  We study the generic dimension of the zero set over $\mathbb{C}^*$,
$\mathbb{R}^*$, and $\mathbb{R}_{>0}$ 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.
