  The value semigroup of a $k$-semiroot $C_k$ of a plane branch $C$ allow us to
recover part of the value semigroup $\Gamma =\langle v_0,\ldots ,v_g\rangle$ of
$C$, that is, it is related to topological invariants of $C$. In this paper we
consider the set of values of differentials $\Lambda_k$ of $C_k$, that is an
analytical invariant, and we show how it determine part of the set of values of
differentials $\Lambda$ of $C$. As a consequence, in a fixed topological class,
we relate the Tjurina number $\tau$ of $C$ with the Tjurina number of $C_k$. In
particular, we show that $\tau\leq \mu-\frac{3n_g-2}{4}\mu_{g-1}$ where
$n_g=gcd(v_0,\ldots ,v_{g-1})$, $\mu$ and $\mu_{g-1}$ denote the Milnor number
of $C$ and $C_{g-1}$ respectively. If $n_g=2$, we have that
$\tau=\mu-\mu_{g-1}$ for any curve in the topological class determined by
$\Gamma$ that is a generalization of a result obtained by Luengo and Pfister.

Algebraic Geometry