On separating fixed points from zero by invariants

TL;DR

The paper proves that in a G-module over a field of positive characteristic, the minimal degree of an invariant separating a fixed point from zero is always a power of p, and provides an explicit method to find such invariants.

Contribution

It demonstrates that the minimal separating invariant degree is a p-power and offers an explicit construction from invariants of degree coprime to p.

Findings

Minimal degree of separating invariants is a p-power.

Explicit method to derive p-power invariants from coprime degree invariants.

Separation of fixed points from zero can be achieved by invariants of degree p^r.

Abstract

Assume a fixed point v in a G-module V can be separated from zero by a homogeneous invariant of degree dp^r where p>0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r , which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.