Ideal theory and algebraic difference equations

Algebraic Difference Equations

On the Algebraic Difference Equations

We continue the study of algebraic difference equations of the type un+2un = ψ(un+1), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in “on some algebraic difference equations un+2un = ψ(un+1) in R∗, related to families of conics or cubics: generalization of the Lyness’ sequences” (2004),...

Algebraic and Algorithmic Aspects of Difference Equations∗

In this course, I will give an elementary introduction to the Galois theory of linear difference equations. This theory shows how to associate a group of matrices with a linear difference equation and shows how group theory can be used to determine properties of the solutions of the equations. I will begin by giving an introduction to the theory of linear algebraic groups, those groups that occ...

Univariate polynomial solutions of algebraic difference equations

Contrary to linear difference equations, there is no general theory of difference equations of the form G(P (x − τ1), . . . , P (x − τs)) + G0(x)=0, with τi ∈ K, G(x1, . . . , xs) ∈ K[x1, . . . , xs] of total degree D ≥ 2 and G0(x) ∈ K[x], where K is a field of characteristic zero. This article is concerned with the following problem: given τi, G and G0, find an upper bound on the degree d of a...

Stability and Bifurcation Analysis of Differential–Difference–Algebraic Equations

This paper treats a nonlinear dynamical system with both continuous-time and discrete-time variables as a differential–difference–algebraic equation (DDA) or a hybrid dynamical system, presents a fundamental analyzing method of such a DDA system for local sampling, asymptotical stability, singular perturbations and bifurcations, and further shows that there exist four types of generic codimensi...