Simultaneous Pairs of Linear and Quadratic Equations In a Galois Field

Galois Theory of Linear Differential Equations

σ-Galois theory of linear difference equations

Inspired by the numerous applications of the differential algebraic independence results from [36], we develop a Galois theory with an action of an endomorphism σ for systems of linear difference equations of the form φ(y) = Ay , where A ∈ GLn(K ) and K is a φσ-field, that is, a field with two given commuting endomorphisms φ and σ, like in Example 2.1. This provides a technique to test whether ...

Galois Theory of Linear Differential Equations - ReadingSample

Simultaneous Integer Values of Pairs of Quadratic Forms

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent “almost all” pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular ...

Optimal Linear Filtering over a Galois Field: Equalization and Prediction

The problem of optimal linear filtering and prediction has been so far typically formulated and studied in the context of realor complex-valued signals. In this article, we provide an extension of this problem to the framework of finite (Galois) fields. Simulation results encompassing supervised and unsupervised prediction-based equalization are presented for a number of scenarios based on GF(2).