Improved convergence estimates for the Schröder-Siegel problem

We reconsider the Schröder-Siegel problem of conjugating an analytic map in {Mathematical expression} in the neighborhood of a fixed point to its linear part, extending it to the case of dimension {Mathematical expression}. Assuming a condition which is equivalent to Bruno's one on the eigenvalues {Mathematical expression} of the linear part, we show that the convergence radius {Mathematical expression} of the conjugating transformation satisfies {Mathematical expression} with {Mathematical expression} characterizing the eigenvalues {Mathematical expression}, a constant {Mathematical expression} not depending on {Mathematical expression} and {Mathematical expression}. This improves the previous results for {Mathematical expression}, where the known proofs give {Mathematical expression}. We also recall that {Mathematical expression} is known to be the optimal value for {Mathematical expression}.