  One considers a system on $\mathbb{C}^2$ close to an invariant curve which
can be viewed as a generalization of the semi-standard map to a trigonometric
polynomial with many Fourier modes. The radius of convergence of an analytic
linearization of the system around the invariant curve is bounded from below by
$\exp(-\frac{2}{d}B(d\alpha)-C)$, where $C\geq 0$ does not depend on $\alpha$,
$d\in \mathbb{N}^*$ and $\alpha$ is the frequency of the linear part. For a
class of trigonometric polynomials, it is also bounded from above by a similar
function. The error function is non decreasing with respect to the smallest
coefficient of the trigonometric polynomial.
