Related papers: Finite Field Experiments (with an Appendix by Stef…
We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.
This paper proposes an efficient FDTD technique for determining electromagnetic fields interacting with a finite-sized 2D and 3D periodic structures. The technique combines periodic boundary conditions---modelling fields away from the edges…
We investigate the problem of how to obtain the force field between atoms of an experimentally determined structure. We show how this problem can be efficiently solved, even at finite temperature, where the position of the atoms differs…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…
This talk reports on the status of an approach to the numerical study of isolated systems with the conformal field equations. We first describe the algorithms used in a code which has been developed at AEI in the last years, and discuss a…
Techniques of zero-temperature field theory that have found application in the analysis of field theory at finite temperature are revisited. Specifically, several of the results that are discussed are relevant to the study of…
Semidefinite programs can be constructed to provide a non-perturbative view of the zero-temperature behavior of quantum systems. This paper examines the properties of these semidefinite programs when applied to lattice-regulated field…
We present new classes of permutation polynomials over finite fields.
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects, i.e. via the finite quotients of the group. But how much understanding can…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…
In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed dimension over a given finite field. We show how this method works for the explicit example of $2$-dimensional algebras over the…
These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in G\"ottingen, June 2007. The first part gives an overview of the methods. The main new result is the…
These notes started to educate ourselves and to collect some background for our future work, with the hope that perhaps they will be useful to others also. Many if not all results are more or less elementary or available in the literature,…
We give a criterion when a polynomial $x^n-g$ is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic.
The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this…
The article contents suggestions on how to perform the Fast Fourier Transform over Large Finite Fields. The technique is to use the fact that the multiplicative groups of specific prime fields are surprisingly composite.
Finite dimensional solutions to a class of stochastic partial differential equations are obtained extending the differential constraints method for deterministic PDE to the stochastic framework. A geometrical reformulation of the stochastic…
This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including…