Local polar invariants and the Poincaré problem in the dicritical case
The carefully written paper under review is of great interest to anyone studying singularities of plane holomorphic foliations. The review that follows cannot be made more precise without providing the numerous required details that the authors include. \par The authors make a further attempt at a complete explanation of the relation between the degree of a holomorphic foliation on the complex projective plane and the degree/genus of an invariant algebraic branch (i.e., the Poincaré Problem). Notwithstanding the existence of invariant curves of arbitrary degree for (necessarily dicritical) foliations of degree $1$ (e.g., $pxdy-qydx$, in a local chart), it might be possible to find conditions on the foliation providing a finer description of the relation between, say, the structure of the desingularization of the foliation and the multiplicity of the invariant curves at the singularities of the former. \par The authors pursue the path of \textit{local polar invariants}, which has proved fruitful in past research [\textit{N. Corral}, Ann. Inst. Fourier 53, No. 3, 787--814 (2003; Zbl 1032.32019); \textit{F. Cano}, \textit{N. Corral} and \textit{R. Mol} , ``Local polar invariants for plane singular foliations'', Expositiones Mathematicae (to appear)] on plane singular foliations, and prove a collection of relevant results relating those polar invariants (i.e., invariants associated to the generic \textit{polar curve} of a foliation) to other properties of the foliation. By using what they call a ``balanced equation'' of the separatrices of a germ of foliation $\mathcal{F}$, they show (for instance) that $\mathcal{F}$ is a generalized curve if and only if the ``polar excess'' (an arithmetic invariant defined in terms of the polar curve and the separatrices of $\mathcal{F}$) is $0$. They also relate the Gómez-Mont-Seade-Verjovsky index of $\mathcal{F}$ (see [\textit{X. Gómez-Mont} et al., Math. Ann. 291, No. 4, 737--752 (1992; Zbl 0725.32012)]) to that polar excess. Using this result, they can prove a bound for the degree $d_0$ of an algebraic invariant curve of an analytic foliation $\mathcal{F}$ on the complex projective plane in terms of the degree $d$ of $\mathcal{F}$ and the intersection numbers of $S$ and the branches in a balanced equation of the separatrices of $\mathcal{F}$, which in the non-dicritical case recovers Carnicer's bound $d_0\leq d+2$ [\textit{M. M. Carnicer}, Ann. Math. (2) 140, No. 2, 289--294 (1994; Zbl 0821.32026)]. \par Then they study the ``topological boundedness'' of several invariants of a singularity $P$ of $\mathcal{F}$: an invariant $I(\mathcal{F})$ is topologically bounded if there is a function $\psi:\mathbb{N}\rightarrow \mathbb{N}$ such that $I(\mathbb{F})\leq \psi(\mu_0(\mathcal{F}))$, where $\mu_0(\mathcal{F})$ is the Milnor number of the singularity of $\mathcal{F}$ at $P$. They show how if $S$ is a germ of curves formed by the union of the \textit{isolated separatrices} and a collection of non-isolated separatrices with special properties, then the algebraic multiplicity of $S$ and the length of its reduction of singularities are both topologically bounded. This has some consequences related to the Poincaré Problem, which they also show. \par A final (and rather important) result concerns the topological invariance of the multiplicity of the singularity of a plane analytic foliation of the second type [\textit{J.-F. Mattei} and \textit{E. Salem}, ``Modules formels locaux de feuilletages holomorphes'', \url{arXiv:math/0402256}]. Specifically, they prove: \par Theorem. Suppose that $\mathcal{F}_1$ and $\mathcal{F}_2$ are topologically equivalent analytic foliations having only convergent separatrices. Then $\mathcal{F}_1$ is of the second type if and only if $\mathcal{F}_2$ is of the second type. As a consequence, $\mathcal{F}_1$ and $\mathcal{F}_2$ have the same algebraic multiplicity.
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