Related papers: Linear estimate for the number of zeros of Abelian…

An Abelian integral is the integral over the level curves of a Hamiltonian $H$ of an algebraic form $\omega$. The infinitesimal Hilbert sixteenth problem calls for the study of the number of zeros of Abelian integrals in terms of the…

The paper deals with the {\it infinitesimal Hilbert 16th problem}: to find an upper estimate of the number of zeros of an Abelian integral regarded as a function of a parameter. In more details, consider a real polynomial $ H$ of degree $…

We give a uniform asymptotic bound for the number of zeros of complete Abelian integrals in domains bounded away from infinity and the singularities.

We study the analogue of the infinitesimal 16th Hilbert problem in dimension zero. Lower and upper bounds for the number of the zeros of the corresponding Abelian integrals (which are algebraic functions) are found. We study the relation…

Let $I(t)= \oint_{\delta(t)} \omega$ be an Abelian integral, where $H=y^2-x^{n+1}+P(x)$ is a hyperelliptic polynomial of Morse type, $\delta(t)$ a horizontal family of cycles in the curves $\{H=t\}$, and $\omega$ a polynomial 1-form in the…

Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $\delta(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form $$I(h)=\int_{\delta(h)} \frac{(\alpha_0+\alpha_1 x+... +…

Consider the vector field $x'= -yG(x, y), y'=xG(x, y),$ where the set of critical points $\{G(x, y) = 0\}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with…

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$.…

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all…

Let $\Delta$ be a root system with a subset of positive roots, $\Delta^+$. We consider edges of the Hasse diagrams of some posets associated with $\Delta^+$. For each edge one naturally defines its type, and we study the partition of the…

Let $\b$ be a Borel subalgebra of a simple Lie algebra $\g$ and let $\Ab$ denote the set of all Abelian ideals of $\b$. We consider $\Ab$ as poset with respect to inclusion, the zero ideal being the unique minimal element of $\Ab$. It was…

In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds…

This paper is devoted to study the limit cycle problem of a cubic reversible system with an isochronous center, when it is perturbed inside a class of polynomials. An upper bound of the number of limit cycles is obtained using the Abelian…

We study the number of rational limit cycles of the Abel equation $x'=A(t)x^3+B(t)x^2$, where $A(t)$ and $B(t)$ are real trigonometric polynomials. We show that this number is at most the degree of $A(t)$ plus one.

We study the maximal number of pairwise distinct columns in a $\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\Delta$. We complement this and…

We prove that the number of limit cycles generated by a small non-conservative 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…

Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…

Given a compact subset $\Sigma$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability…