An explicit formula for period determinant

Glutsyuk, Alexey

doi:10.5802/aif.2204

Cited by 5 publications

(9 citation statements)

References 8 publications

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“…a i are the critical values of H ( [2], see [3] for more details, where an explicit formula for the constant C(h, Ω) was obtained). The proof of the lower bound of ∆(t) is based on the latter formula.…”

Section: Remarkmentioning

confidence: 99%

“…for the period determinant, see [3] and [2], which holds, if the form tuple under consideration is d-standard, see the following Definition. 1.54 Example Let (l(i), m(i)) be a lexicographic sequence of pairs (l, m), 0 ≤ l, m ≤ n − 1, numerated by i = 1, .…”

Section: )mentioning

confidence: 99%

“…The proof of this statement, which is omitted to save the space, can be easily derived from the results of this Subsection and Section 3. In general, one can choose a generic h in such a way that the determinant ∆ h,Ω (t) constructed with the above ω i be identically equal to 0 (this follows from results of [3]): in this case the form tuple should be changed.…”

Section: )mentioning

confidence: 99%

“…where P d are the polynomials from [3], whose definition is recalled below, Now all the entries of formula (1.28) are defined. As it will be shown below, Theorem 1.52 is implied by the following 1.57 Theorem Let h be a given generic polynomial of degree n+ 1 ≥ 3 with ||h|| max = 1.…”

Section: )mentioning

confidence: 99%

“…(2.41) This follows from inequalities |x − x i | ≥ ε (which hold by assumption), (2.40) and (2.10). To show that Σ t (x) = 0 for small t, we estimate from above the derivative of Σ t (x) in t by using the following a priori upper bound of Σ t (x) valid whenever |t| ≤ 5 and |x| ≤ X n : 3 (2.42) (since |y i (x, t)| ≤ Y n and by elementary inequalities). The classical inequality on derivative of holomorphic function (which follows from Cauchy lemma) says that if ψ(t) is a bounded function holomorphic in t ∈ D R , |ψ| < M, then |ψ ′ (t)| ≤ M R−|t| .…”

Section: Noncritical Values Of Projection Of Variable Level Curvementioning

confidence: 99%

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On the number of zeros of Abelian integrals

2010

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Abstract. We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in a small nonconservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing infinitesimal Hilbert 16th problem.The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection defined over Q (the Gauss-Manin connection) with a quasiunipotent monodromy group.

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Lectures on Analytic Differential Equations

Ilyashenko

Yakovenko

2007

Graduate Studies in Mathematics

314

593

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The book combines the features of a graduate-level textbook with those of a research monograph and survey of the recent results on analysis and geometry of differential equations in the real and complex domain. As a graduate textbook, it includes self-contained, sometimes considerably simplified demonstrations of several fundamental results, which previously appeared only in journal publications (desingularization of planar analytic vector fields, existence of analytic separatrices, positive and negative results on the Riemann-Hilbert problem, Ecalle-Voronin and Martinet-Ramis moduli, solution of the Poincaré problem on the degree of an algebraic separatrix, etc.). As a research monograph, it explores in a systematic way the algebraic decidability of local classification problems, rigidity of holomorphic foliations, etc. Each section ends with a collection of problems, partly intended to help the reader to gain understanding and experience with the material, partly drafting demonstrations of the more recent results surveyed in the text.The exposition of the book is mostly geometric, though the algebraic side of the constructions is also prominently featured. On several occasions the reader is introduced to adjacent areas, such as intersection theory for divisors on the projective plane or geometric theory of holomorphic vector bundles with meromorphic connections. The book provides the reader with the principal tools of the modern theory of analytic differential equations and intends to serve as a standard source for references in this area. ReadershipGraduate students and research mathematicians interested in analysis and geometry of differential equations in real and complex domain. Ordering information: Visit the AMS online bookstore Draft version is available onlineThe draft posted on this site is really outdated, with many inaccuracies, sloppy phrases and even errors. Besides, a number of sections had been added or modified. It sole purpose is to give an idea what appears in the printed book.Do not quote this draft to avoid misleading enumeration of pages, theorems and definitions. Contents PrefaceThe branch of mathematics which deals with ordinary differential equations can be roughly divided into two large parts, qualitative theory of differential equations and the dynamical systems theory. The former mostly deals with systems of differential equations on the plane, the latter concerns multidimensional systems (diffeomorphisms on two-dimensional manifolds and flows in dimension greater than two and up to infinity). The former can be considered as a relatively orderly world, while the latter is the realm of chaos.A key problem, in some sense a paradigm influencing the development of dynamical systems theory from its origins, is the problem of turbulence: how a deterministic nature of a dynamical system can be compatible with its apparently chaotic behavior. This problem was studied by the precursors and founding fathers of the dynamical systems theory: L. Landau, H. Hopf, A. Kolmogorov, V. Arnold, S. Smal...

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An explicit formula for period determinant

Cited by 5 publications

References 8 publications

Upper Bounds of Topology of Complex Polynomials in Two Variables

Upper Bounds of Topology of Complex Polynomials in Two Variables

On the number of zeros of Abelian integrals

Lectures on Analytic Differential Equations

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