Algebraic independence of the Carlitz period and its hyperderivatives

TL;DR

This paper proves that the Carlitz period and all its hyperderivatives are algebraically independent over the base field, revealing explicit polynomial relations and connections to tensor powers of the Carlitz module.

Contribution

It establishes the algebraic independence of the Carlitz period and its hyperderivatives, and provides explicit polynomial expressions linking these hyperderivatives to tensor power coordinates.

Findings

Carlitz period and hyperderivatives are algebraically independent

Explicit polynomial expressions relate hyperderivatives to tensor power coordinates

Connections to period lattice generators of tensor powers

Abstract

This paper deals with the fundamental period $\tilde{\pi}$ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field $\mathbb{F}_q(\theta)$. Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in $\tilde{\pi}$ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here.