We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot{z}=$${{z}^{3}}+{{\epsilon }_{1}}z+{{\epsilon }_{0}}$ for $z\in \mathbb{C}{{\mathbb{P}}^{1}}$, depending on the values of ${{\epsilon }_{1}},{{\epsilon }_{0}}\in \mathbb{C}$. The bifurcation diagram is in ${{\mathbb{R}}^{^{4}}}$, but its conic structure allows describing it for parameter values in ${{\mathbb{S}}^{3}}$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

We consider isomonodromic deformations of second-order Fuchsian differential equations on elliptic curves. The isomonodromic deformations are described as a completely integrable Hamiltonian system.