On the number of zeros of Melnikov functions

Sergey Benditkis; Dmitry Novikov

Displaying similar documents to “On the number of zeros of Melnikov functions”

Higher order Poincaré-Pontryagin functions and iterated path integrals

Lubomir Gavrilov (2005)

Annales de la Faculté des sciences de Toulouse : Mathématiques

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Effective bounds for the zeros of linear recurrences in function fields

Clemens Fuchs, Attila Pethő (2005)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable. Moreover, we study similar problems in this context as the equation $G_{n} (x) = G_{m} (P (x)), (m, n) \in ℕ^{2}$ , where $(G_{n} (x))$ is a linear recurring sequence of polynomials and $P (x)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

Eisenstein's theorem on power series expansions of algebraic functions

Wolfgang Schmidt (1990)

Acta Arithmetica

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Solvability of certain equations in a finite field

L Carlitz (1962)

Acta Arithmetica

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Criterion on L-boundedness for a class of oscillatory singular integrals with rough kernels.

Shan Zhen Lu, Yan Zhang (1992)

Revista Matemática Iberoamericana

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The expression of a polynomial as a sum of three irreducibles

D. Hayes (1966)

Acta Arithmetica

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Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

Lubomir Gavrilov (1999)

Annales de l'institut Fourier

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Let $𝒜$ be the real vector space of Abelian integrals $I (h) = \int \int_{{H \leq h}} R (x, y) d x \land d y, h \in [0, \tilde{h}]$ where $H (x, y) = (x^{2} + y^{2}) / 2 + ...$ is a fixed real polynomial, $R (x, y)$ is an arbitrary real polynomial and ${H \leq h}$ , $h \in [0, \tilde{h}]$ , is the interior of the oval of $H$ which surrounds the origin and tends to it as $h \to 0$ . We prove that if $H (x, y)$ is a semiweighted homogeneous polynomial with only Morse critical points, then $𝒜$ is a free finitely generated module over the ring of real polynomials $ℝ [h]$ , and compute its rank. We find the generators of $𝒜$ in the case when $H$ is an...