Algebraic independence and linear difference equations

Algebraic Difference Equations

Sharp algebraic periodicity conditions for linear higher order difference equations

In this paper we give easily verifiable, but sharp (in most cases necessary and sufficient) algebraic conditions for the solutions of systems of higher order linear difference equations to be periodic. The main tool in our investigation is a transformation, recently introduced by the authors, which formulates a given higher order recursion as a first order difference equation in the phase space...

Criteria for irrationality, linear independence, transcendence and algebraic independence

For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coeff...

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),...

Frobenius Difference Equations and Algebraic Independence of Zeta Values in Positive Equal Characteristic

In analogy with the Riemann zeta function at positive integers, for each finite field Fpr with fixed characteristic p we consider Carlitz zeta values ζr(n) at positive integers n. Our theorem asserts that among the zeta values in ∪∞ r=1 {ζr(1), ζr(2), ζr(3), . . . }, all the algebraic relations are those relations within each individual family {ζr(1), ζr(2), ζr(3), . . . }. These are the algebr...