Simple exponential estimate for the number of real zeros of complete abelian integrals

Dmitri Novikov; Sergei Yakovenko

Displaying similar documents to “Simple exponential estimate for the number of real zeros of complete abelian integrals”

Analytic continuation of fundamental solutions to differential equations with constant coefficients

Christer O. Kiselman (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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If $P$ is a polynomial in $R^{n}$ such that $1 / P$ integrable, then the inverse Fourier transform of $1 / P$ is a fundamental solution $E_{P}$ to the differential operator $P (D)$ . The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$ . For $n = 1$ it is shown that $E_{P}$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$ . The singularities of this extension are studied.

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

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Let p(z) be a polynomial of the form $p (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ , $a_{j} \in - 1, 1$ . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

On distance between zeros of solutions of third order differential equations

N. Parhi, S. Panigrahi (2001)

Annales Polonici Mathematici

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The lower bounds of the spacings b-a or a’-a of two consecutive zeros or three consecutive zeros of solutions of third order differential equations of the form y”’ + q(t)y’ + p(t)y = 0 (*) are derived under very general assumptions on p and q. These results are then used to show that $t_{n + 1} - t ₙ \to \infty$ or $t_{n + 2} - t ₙ \to \infty$ as n → ∞ under suitable assumptions on p and q, where ⟨tₙ⟩ is a sequence of zeros of an oscillatory solution of (*). The Opial-type inequalities are used to derive lower bounds of the spacings d-a...

The Lehmer constants of an annulus

Artūras Dubickas, Chris J. Smyth (2001)

Journal de théorie des nombres de Bordeaux

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Let $M (α)$ be the Mahler measure of an algebraic number $α$ , and $V$ be an open subset of $ℂ$ . Then its $L (V)$ is inf $M {(α)}^{1 / deg (α)}$ , the infimum being over all non-zero non-cyclotomic $α$ lying with its conjugates outside $V$ . We evaluate $L (V)$ when $V$ is any annulus centered at $0$ . We do the same for a variant of $L (V)$ , which we call the transfinite Lehmer constant $L_{\infty} (V)$ .Also, we prove the converse to Langevin’s Theorem, which states that $L (V) > 1$ if $V$ contains a point of modulus $1$ . We prove the corresponding result for...

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...

Zeros of derivatives of solutions to singular $(p, n - p)$ conjugate BVPs

Irena Rachůnková, Staněk, Svatoslav (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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Positive solutions of the singular $(p, n - p)$ conjugate BVP are studied. The set of all zeros of their derivatives up to order $n - 1$ is described. By means of this, estimates from below of the solutions and the absolute values of their derivatives up to order $n - 1$ on the considered interval are reached. Such estimates are necessary for the application of the general existence principle to the BVP under consideration.

Simple zeros of degree 2 $L$ -functions

Andrew R. Booker (2016)

Journal of the European Mathematical Society

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We prove that the complete $L$ -functions of classical holomorphic newforms have infinitely many simple zeros.

Zeros of a certain class of Gauss hypergeometric polynomials

Addisalem Abathun, Rikard Bøgvad (2018)

Czechoslovak Mathematical Journal

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We prove that as $n \to \infty$ , the zeros of the polynomial $_{2} F_{1} [\begin{matrix} - n, α n + 1 \\ α n + 2 \end{matrix}; z]$ cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999–2001) to the case of a complex parameter $α$ and partially proves a conjecture made by the authors in an earlier work.

Some Results on the Properties of Differential Polynomials Generated by Solutionsof Complex Differential Equations

Zinelâabidine LATREUCH, Benharrat BELAÏDI (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation $f^{(k)} + A_{k - 1} (z) f^{(k - 1)} + \dots + A_{1} (z) f^{'} + A_{0} (z) f = 0,$ where $A_{i} (z)$ $(i = 0, 1, \dots, k - 1)$ are meromorphic functions of finite order in the complex plane.

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.