Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic

TL;DR

This paper proves that the algebraic relations among Carlitz zeta values over finite fields are precisely those generated within each family, using a new motivic method for Frobenius difference equations.

Contribution

It introduces a motivic approach to establish algebraic independence of Carlitz zeta values in positive characteristic.

Findings

Algebraic relations are confined within individual zeta value families.

Develops a new method for analyzing Frobenius difference equations.

Establishes algebraic independence results for Carlitz zeta values.

Abstract

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