Zero-curry curve equals coefficient-line zero set of its constant term

Let $h$ be a bivariate real polynomial with $h_{deg0} : (\mathrm{Curry0}\,h).\mathrm{natDegree} = 0$, i.e. $h$ viewed as a univariate polynomial in the first variable (over the coefficient ring $\mathrm{XCoeff}$) is constant. Then its plane zero set is exactly the coefficient-line zero set of that constant coefficient:

This characterizes degenerate (vertical) curves: a polynomial independent of the first variable cuts out a union of vertical lines $X_1 = x$ indexed by the roots of its coefficient. It is the structural lemma behind every zero-curry case split, reducing intersections involving such an $h$ to fiberwise univariate root counts.