Related papers: Introducing an Analysis in Finite Fields

Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension…

A table of sums useful for generating function applications (discrete Laplace transforms or z-transforms). Related definitions and formulas (including Lagrange's expansion), and reference to formulas in Abramowitz and Stegun, Handbook of…

We derive some new finite sums involving the sequence $s_{2}\left(n\right),$ the sum of digits of the expansion of $n$ in base $2.$ These functions allow us to generalize some classical results obtained by Allouche, Shallit and others.

We give an a geometric interpretation of the Hasse-Arf theorem for function fields using the recently proved Oort conjecture.

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no…

The present article is an extended version of [6] containing new results and an updated list of references. We review the notion of polar analyticity introduced in a previous paper and succesfully applied in Mellin analysis and quadrature…

We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the…

Physically relevant field-theoretic quantities are usually derived from perturbation techniques. These quantities are solved in the form of an asymptotic series in powers of small perturbation parameters related to the physical system, and…

In this paper we present a finite field analogue for one of the Appell series. We shall derive its transformations, reduction formulas as well as generating functions.

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

Let D be a valued division algebra, finite-dimensional over its center F. Assume D has an unramified splitting field. The paper shows that if D contains a maximal subfield which is Galois over F (i.e. D is a crossed product) then the…

Plan of this report is given below: 1. Motivation from Physical and Mathematical Point of View; 2. Differential Calculi on Finite Groups; 3. Metrics; 4. Lagrangian Field Theory and Symplectic Structure; 5. Scalar Field Theory and Spectral…

We prove that infinite Galois extensions of number fields with Galois group of finite exponent have the Northcott property. The main novelty of our approach lies in the application of a theorem of Segal on profinite groups.

We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is…

Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…

Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…

We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115}…

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…

Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for…

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and…