Related papers: Dynamical and qKZ equations modulo $p^s$, an examp…

We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.

We consider the KZ equations over $\mathbb C$ in the case, when the hypergeometric solutions are hyperelliptic integrals of genus $g$. Then the space of solutions is a $2g$-dimensional complex vector space. We also consider the same…

In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an…

Let p be a prime number, and let k be an imaginary quadratic field in which p decomposes into two primes \mathfrak{p} and \bar{\mathfrak{p}}. Let k_\infty be the unique Z_p-extension of k which is unramified outside of \mathfrak{p}, and let…

We review elliptic solutions to integrable nonlinear partial differential and difference equations (KP, mKP, BKP, Toda) and derive equations of motion for poles of the solutions. The pole dynamics is given by an integrable many-body system…

This paper has two main goals. The first is universality of the KPZ equation for fluctuations of dynamic interfaces associated to interacting particle systems in the presence of open boundary. We consider generalizations on the open-ASEP…

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…

For a second-order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of…

The KZ equations are differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex…

The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters. It is achieved by considering the…

We show that the existence of a non-trivial solution of $x^n+y^n=p^n$, with $p$ a prime number, is equivalent to the existence of a solution of a certain (over-determined) system of $(n-1)$-recursion relations ("zipper" equations) in…

Let p\in\{2,3\}, and let k be an imaginary quadratic field in which p decomposes into two distinct primes \mathfrak{p} and \bar{\mathfrak{p}}. Let k_\infty be the unique Z_p-extension of k which is unramified outside of \mathfrak{p}, and…

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…

We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…

Using the $3D$ mirror symmetry we construct a system of polynomials $T_s(z)$ with integral coefficients which solve the quantum differential equitation of $X=T^{*} Gr(k,n)$ modulo $p^s$, where $p$ is a prime number. We show that the…

We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions.

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $\mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for…

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv…

We prove the height functions for a class of non-integrable and non-stationary particle systems converge to the KPZ equation, thereby making progress on the universality of the KPZ equation. The models herein are ASEP [4] with a mesoscopic…

It is known that solutions of the KZ equations can be written in the form of multidimensional hypergeometric integrals. In 2017 in a joint paper of the author with V. Schechtman the construction of hypergeometric solutions was modified, and…