Improved convergence estimates for the Schroder-Siegel problem

We reconsider the Schroder-Siegel problem of conjugating an analytic map in $mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Brunos one on the eigenvalues $lambda_1,ldots,lambda_n$ of the linear part we show that the convergence radius $rho$ of the conjugating transformation satisfies $ln rho(lambda )geq -CGamma(lambda)+C$ with $Gamma(lambda)$ characterizing the eigenvalues $lambda$, a constant $C$ not depending on $lambda$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.