8 problems

Let ${\cal A}_\mu$ and ${\cal A}_e$ be the vector spaces of Abelian integrals $\alpha_0 I_0+\alpha_1 I_1$ corresponding to the continuous families of ovals ${\cal O}_\mu$ and…

Let $F(h)=I_1(h)/I_0(h)$, where $I_0$ and $I_1$ are the Abelian integrals associated with the family of ovals, and let $\lambda\in\mathbb{C}$ be the parameter. Consider the equatio…

Let $P(x)$ be a polynomial with distinct real critical values, and let ${\cal D}$ be the relevant domain of the parameter $h$. Consider the vector space of Abelian integrals…

Let $F(x,\lambda)$ be the polynomial family and let $C_0(\lambda)$ be the matrix defining the Pfaffian Picard–Fuchs system. Write … A rational $1$-form has a logarithmic singularit…

Let $n$ be a positive integer. There are constants $\delta(n)$, $l(n)$, and $R(n)$ such that the following holds. Let $H_0$ be an arbitrary real polynomial in $\mathcal H_n$, let…

Arnol'd's conjecture. Abelian integrals corresponding to natural problems form a Chebyshev system.

Constructive bound conjecture. There exists an explicit (constructive) bound of the form

Gautier–Gavrilov–Iliev conjecture. The cyclicity of these period annuli is two, except in particular cases in which it is three.