= Uniform analyticity under polynomial coordinate changes
{title2=$\|\partial^\beta f_\varepsilon\|_\infty\le A\beta!R^{-|\beta|}$}
Suppose a family of <polynomial> coordinate maps has a fixed bound on its degrees and coefficients depending continuously on a compact parameter set, and sends a fixed real compact set into a compact subset of the domain of a <real analytic function>. A common complex neighborhood can then be chosen so the compositions are holomorphic and uniformly bounded there. The multivariable <Cauchy estimate> gives uniform factorial <derivative> bounds. For the change $\tilde x=x+y^2-\varepsilon$, the coefficient pullback uses its inverse: $f_\varepsilon(\tilde x,\tilde y)=f(\tilde x-\tilde y^2+\varepsilon,\tilde y)$.
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