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 algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.