  In this note, we present some results on the Mattei's conjecture, which
asserts that the algebraic multiplicity of one-dimensional holomorphic
foliations is a topological invariant. For instance, we establish a version of
the A'Campo-L\^e Theorem adapted for foliations, which was originally
formulated for hypersurfaces. More precisely, we show that, under certain
conditions, the algebraic multiplicity equals to 1 of holomorphic
one-dimensional foliations is a topological invariant. As a consequence, we
prove that saddle-nodes are also topological invariant. Additionally, we
address a fundamental result in the literature concerning Mattei's conjecture
proved by Rosas-Bazan, which says that under the existence of a homeomorphism
extending throughout a neighborhood of the exceptional divisor of the first
blowing up, Mattei's conjecture has a positive answer. As a consequence of a
characterization of the invariance of multiplicity proved here, we obtain that
Mattei's conjecture has a positive answer if the homeomorphism extends locally
in a small neighborhood where there is a separatrix not contained in the
exceptional divisor of the first blowing up, but not necessarily extends
throughout a neighborhood of whole exceptional divisor.
