Related papers: Determinant of $\mathbb F_p$-hypergeometric soluti…

It is shown how to define difference equations on particular lattices $\{x_n\}$, $n\in\mathbb{Z}$, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special…

Thanks to earlier work of Koiran, it is known that the truth of the Generalized Riemann Hypothesis (GRH) implies that the dimension of algebraic sets over the complex numbers can be determined within the polynomial-hierarchy. The truth of…

Let $p$ be a prime and $n$ a positive integer. As the first main result, we present a deterministic algorithm for deciding whether the matrix algebra $\mathbb{F}_p[A_1,\dots,A_t]$ with $A_1,\dots,A_t \in \mathrm{GL}(n,\mathbb{F}_p)$ is a…

One deals with degenerations by coordinate sections of the square generic Hankel matrix over a field $k$ of characteristic zero, along with its main related structures, such as the determinant of the matrix, the ideal generated by its…

For a large prime $p$, and a polynomial $f$ over a finite field $F_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of $F_p^*$ containing $H\ge 1$ consecutive values $f(x)$, $x = u+1, \ldots, u+H$,…

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to…

The solutions of the equation $f^{(p-1)} + f^p = h^p$ in the unknown function $f $over an algebraic function field of characteristic $p$ are very closely linked to the structure and factorisations of linear differential operators with…

Let $E_p(x)$ denote the Artin-Hasse exponential and let $\overline{E}_p(x)$ denote its reduction modulo $p$ in $\mathbb{F}_p[[x]]$. In this article we study transcendence properties of $\overline{E}_p(x)$ over $\mathbb{F}_p[x]$. We give two…

Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.

In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an…

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…

In this paper we shed more light on determinants of interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both…

We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we…

For any polynomial $p$ on $\mathbf{C}^{n}$, a variety $ V_{p} = \{z \in \mathbf{C}^{n} ; p(z)=0 \} $ will be considered. Let $\text{Exp}(V_{p})$ be the space of holomorphic functions of expotential growth on $V_{p}$. We shall prove that the…

In the total matching problem, one is given a graph $G$ with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the…

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…

A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original…

The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from $q=e^{2\pi i \tau}$ to the reciprocal of the…

Let F : V --> V be a self-morphism of a quasiprojective variety defined over a number field K and let P be a point in V(K) with infinite orbit under iteration of F. For each prime ideal p of good reduction, let m_p(F,P) be the size of the…