On the finite generation of additive group invariants in positive characteristic

TL;DR

This paper demonstrates that, unlike in characteristic zero, the invariants of additive group actions in positive characteristic are finitely generated, providing characteristic-free analogs of known counterexamples.

Contribution

It establishes that in positive characteristic, the invariants are finitely generated, contrasting with known non-finite generation examples in characteristic zero.

Findings

Invariants are finitely generated in positive characteristic.

Characteristic-free analogs of counterexamples are constructed.

Contrasts with characteristic zero cases are clarified.

Abstract

Roberts, Freudenburg, and Daigle and Freudenburg have given the smallest counterexamples to Hilbert's fourteenth problem as rings of invariants of algebraic groups. Each is of an action of the additive group on a finite dimensional vector space over a field of characteristic zero, and thus, each is the kernel of a locally nilpotent derivation. In positive characteristic, additive group actions correspond to locally finite iterative higher derivations. We set up characteristic-free analogs of the three examples, and show that, contrary to characteristic zero, in every positive charateristic, the invariants are finitely generated.