Related papers: Solutions of the $sl_2$ qKZ equations modulo an in…
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…
We consider the quantized Knizhnik-Zamolodchikov difference equation (qKZ) with values in a tensor product of irreducible sl(2) modules, the equation defined in terms of rational R-matrices. We solve the equation in terms of…
We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions.
We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the elliptic quantum group $E_{\tau,\eta}(sl_2)$, the equation…
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…
We consider the KZ differential equations over $\mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We…
We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.
We consider the $sl(2)$ quantized Knizhnik-Zamolodchikov equation (qKZ), defined in terms of rational R-matrices. The properties of the equation change when the step of the equation takes a resonance value. In this case the discrete…
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…
We construct the hypergeometric solutions for the quantized KZ equation with values in a tensor product of vector representations of $U_q(sl_n)$ at $|q|=1$ and give an explicit formula for the corresponding determinant in terms of the…
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ Verma modules. We solve the equation in terms…
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group S_N is presented for integer values of the parameter. The corresponding integrals…
The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $\zeta_{\beta}(z)$ of the $\beta$-shift, for $\beta > 1$ close to one, is investigated. We briefly recall how this family is…
The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the…
Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\…
Consider polynomials $F_1,\dots,F_s$ in $\K[X_1,\dots,X_n]$ over a field $\K$, their zero-set $V(F_1,\dots,F_n)$ in $\Kbar^n$ and its decomposition into equidimensional components $V_0,\dots,V_n$ (with $V_i$ either empty or of dimension $i$…
We study multivariate monomial Vandermonde matrices $V_N(Z)$ with arbitrary distinct nodes $Z=\{z_1,\dots,z_s\}\subset B_2^n$ in the high-degree regime $N\ge s-1$. Introducing a projection-based geometric statistic -- the \emph{max-min…
We construct integral representations of solutions to the boundary quantum Knizhnik-Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine $\mathfrak{sl}_2$, with the…
A difference equation analogue of the Knizhnik-Zamolodchikov equation is exhibited by developing a theory of the generating function $H(z)$ of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the…