### ?

## Bounded limit cycles of polynomial foliations of ℂ²

Bulletin of the Brazilian Mathematical Society. 2017. No. 1.

Kudryashov Y., Goncharuk N. B.

In press

In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.

Keywords: монодромияпредельный циклголоморфное слоениеComplex limit cyclescomplex foliationsmonodromy group

Publication based on the results of:

Serge Lvovski, Manuscripta Mathematica 2014 Vol. 145 P. 235-242

We show that using an idea from a paper by Van de Ven one may obtain a
simple proof of Zak's classification of smooth projective surfaces
with zero vanishing cycles. This method of proof allows one to extend
Zak's theorem to the case of finite characteristic. ...

Added: October 14, 2014

V. Kruglov, O. Pochinka, G. Talanova, Proceedings of the International Geometry Center 2020 Vol. 13 No. 1 P. 49-60

Currently, a complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9,7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of ...

Added: June 28, 2020

Vyugin I. V., Гонцов Р. Р., Успехи математических наук 2012 Т. 67 № 3 (405) С. 183-184

Получено обобщение результата Ильяшенко-Хованского, утверждающего, что разрешимость в квадратурах фуксовой системы с малыми коэффициентами эквивалентна ее треугольности. В работе этот результат обобщен на случай систем с малыми собственными значениями матриц вычетов. ...

Added: February 21, 2013

S.A.Chistyakova, Dolov M. V., Differential Equations 2012 Vol. 48 No. 8 P. 1180-1182

For a certain class of two-dimensional autonomous systems of differential equations with an invariant curve that contains ovals, we indicate necessary and sufficient conditions for these ovals to be limit cycles of phase trajectories. ...

Added: March 15, 2013

Esterov A. I., Gusev G. G., Mathematische Annalen 2016 Vol. 365 No. 3 P. 1091-1110

We generalize the Abel–Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued ...

Added: February 27, 2017

Lvovsky S., / Cornell University. Series math "arxiv.org". 2013. No. 1305.2205.

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic. ...

Added: October 3, 2013

Vyugin I. V., Успехи математических наук 2011 Т. 66 № 1 (397) С. 37-64

Работа посвящена проблеме Римана–Гильберта для скалярных фуксовых уравнений: задаче построения скалярного фуксова уравнения по представлению монодромии и набору особых точек. Основную часть работы представляют результаты А. А. Болибруха [5], М. Ван-дер-Пута и М. Зингера [7] и автора [10], обобщенные в единую теорему, снабженную новым доказательством. Обсуждаются также некоторые из возможных приложений этих результатов.
Библиография: 16 названий ...

Added: February 27, 2013

Esterov A. I., Takeuchi K., Ando K., Advances in Mathematics 2015 Vol. 272 P. 1-19

We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson. In particular, we compute the monodromy zeta-function. ...

Added: October 10, 2014

Esterov A. I., Takeuchi K., Lemahieu A., Journal of the European Mathematical Society 2021

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations --- the Denef--Loeser conjecture on topological zeta functions --- is open for surface singularities. We prove it for a wide ...

Added: November 28, 2020

Ilyashenko Y., Publicacions Matematiques 2014 Vol. EXTRA P. 279-286

Complex limit cycle located in a neighborhood of a hyperbolic polycycle can not vanish under a small deformation that preserves the characteristic values of the vertexes of the polycycle. The cycles either change holomorphically under the change of the parameter, or come to the boundary of the fixed neighborhood of the polycycle. The present paper ...

Added: March 11, 2015

Takeuchi K., Esterov A. I., Lemahieu A., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1309.0630v4.

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain higher-dimensional analogues of their results, which, in particular, prove the conjecture for all isolated singularities of 4 variables, as well as for many classes of non-isolated and higher-dimensional singularities. One ...

Added: September 18, 2017

Brav C. I., Thomas H., Compositio Mathematica 2014 Vol. 150 No. 3 P. 343-333

We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. ...

Added: September 29, 2014

Kolobianina A., Kruglov V., Журнал Средневолжского математического общества 2020 Т. 22 № 4 С. 434-441

In this paper, we consider the class of Ω-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of Ω-stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of ...

Added: November 27, 2020

Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597-613

We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...

Added: August 30, 2019

Esterov A. I., Compositio Mathematica 2019 Vol. 155 No. 2 P. 229-245

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.
In particular, our result proves the multivariate ...

Added: February 5, 2019

Serge Lvovski, / Cornell University. Series arXiv "math". 2017.

We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of ...

Added: June 14, 2017

Goncharuk N. B., Kudryashov Y., / Cornell University. Series math "arxiv.org". 2015. No. 1504.03313.

In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain. ...

Added: April 15, 2015

V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Campana F., Demailly J., Verbitsky M., Algebraic Geometry 2014 Vol. 2 P. 131-139

We prove that any compact Kahler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classication of compact Kahler manifolds. The proof follows from the Brunella pseudo-eectivity theorem, combined with fundamental results ...

Added: April 29, 2014

Kondratieva L. A., A.V. Romanov, Electronic Journal of Qualitative Theory of Differential Equations 2019 No. 96 P. 1-11

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincaré–Bendixson theory. In the case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of ...

Added: December 22, 2019

A.V. Romanov, Kondratieva L. A., / Cornell University. Series math "arxiv.org". 2019. No. 1911.03932.

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small ...

Added: November 13, 2019

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Yu. Burman, Serge Lvovski, Moscow Mathematical Journal 2015 Vol. 15 No. 1 P. 31-48

Suppose that C ⊂ P^2 is a general enough nodal plane curve
of degree > 2, : \hat C → C is its normalization, and π: C′ → P^1 is a finite
morphism simply ramified over the same set of points as a projection
pr_p ◦ν : \hat C → P1, where p ∈ P^2 ...

Added: January 14, 2015

A.V. Romanov, Kondratieva L. A., / Cornell University. Series math "arxiv.org". 2019. No. 1911.03932.

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small ...

Added: November 13, 2019