= Solution
Fix the coefficient normal form and a positive analyticity exponent $\zeta_0$ once and for all. If $g$ is a <polynomial>, its pullback $\tilde g_\varepsilon=g(\tilde x-\tilde y^2+\varepsilon,\tilde y)$ is again a <polynomial>, with degree bounded independently of $\varepsilon$. On a fixed closed coordinate ball, its finitely many nonzero <derivatives> are uniformly bounded as the parameter ranges over a fixed compact small interval. Hence there is a finite $A_g\ge1$ such that
$$
\|\partial^\beta(\tilde g_\varepsilon/A_g)\|_\infty\le\beta!\zeta_0^{|\beta|}\quad\text{for every }\beta\text{ and every small }\varepsilon.
$$
All higher <derivatives> vanish, so arbitrarily high <polynomial> degree causes no failure of this bound; it only changes $A_g$.
The normalized forcing belongs to one common majorant class. The <Cauchy-Kovalevskaya theorem> therefore gives a radius $r_*>0$ independent of $g$. The problem is linear with zero data, so multiplying the normalized solution by $A_g$ solves the original forcing without changing its radius. Choose one $\varepsilon$ with the cap inside this radius. Thus
$$
\boxed{\text{one }\varepsilon>0\text{ works for every polynomial }g.}
$$
This is the <uniform Cauchy radius for polynomial forcing>. It asserts a common domain, not a uniform solution-amplitude bound over all <polynomials>.
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