Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1608.07217

The Poincar\'e problem in the dicritical case

Yohann Genzmer·Rogério Mol
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the $GSV$-index. We apply it to the Poincar\'e problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincar\'e problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, ${\rm deg}(S) \leq {\rm deg}(\mathcal{F}) + 2$.

Explore related subjects

math.DSmath.CV

Keep this discovery

BibTeXRIS

Yohann Genzmer, Rogério Mol. 2016-08-25. The Poincar\'e problem in the dicritical case. https://arxiv.org/abs/1608.07217

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗