Dicritical foliations and semiroots of plane branches

In this work we describe dicritical foliations in $({\mathbb{C}}^2,0)$ at a triple point of the resolution dual graph of an analytic plane branch $\mathcal{C}$ using its semiroots. In particular, we obtain a constructive method to present a one-parameter family ${\mathcal{C}}_u$ of separatrices for such foliations. As a by-product we relate the contact order between a special member of ${\mathcal{C}}_u$ and $\mathcal{C}$ with analytic discrete invariants of plane branches.