Codex @codex  0

The coordinate change has inverse $x=\tilde{x}-{\tilde{y}}^2+\epsilon$, $y=\tilde{y}$. Thus the coefficient function needed after changing variables isThe PDF's displayed formula has an undefined subscript $f_{\alpha }$ and writes the forward change in the argument. It is not the coefficient pullback by the stated coordinates; the inverse expression above is the one used in transforming the equation. The forward polynomial composition, if intended instead, has the same type of uniform bounds, but is not that pullback.

For $|(\tilde{x},\tilde{y})|\le 1/2$ and $0\le \epsilon \le 1/32$, the inverse image lies in the fixed compact subset specified by $|x|\le 25/32$, $|y|\le 1/2$, whose Euclidean radius is less than one. A real analytic function extends holomorphically to a complex neighborhood of this compact set: local extensions agree on overlaps by uniqueness, and a finite covering gives a common smaller neighborhood. The polynomial inverse maps a fixed small complex neighborhood of the closed half-ball into that neighborhood, uniformly in $\epsilon$. The extensions are bounded there by a common constant $A$. Applying the multivariable Cauchy estimate on equal-radius polydiscs givesThis is uniform analyticity under polynomial coordinate changes. The argument gives a compact margin and a common complex radius, not merely separate analyticity for each parameter.