Uniform Cauchy radius for polynomial forcing

For a linear real analytic non-characteristic Cauchy problem for a partial differential equation with zero data, the local real analytic radius may be chosen uniformly over polynomial forcing terms. Fix an analyticity exponent $R^{-1}$. Each polynomial, including a polynomial obtained by a polynomial coordinate change, satisfies the corresponding factorial derivative bound with some finite prefactor, uniformly over a compact parameter range. Divide the forcing by that prefactor to put all such data in one fixed majorant class. The Cauchy-Kovalevskaya theorem supplies a common radius, and linearity rescales the solution back without shrinking it. This argument does not bound the amplitudes of the resulting solutions uniformly.