Separating polynomial invariants over non-closed fields of finite abelian groups

TL;DR

This paper extends the understanding of polynomial invariants that distinguish orbits in finite abelian group representations, especially over non-algebraically closed fields like rationals or reals, with degree bounds up to 3.

Contribution

It generalizes degree bounds for separating invariants from algebraically closed fields to non-closed fields such as rationals and reals.

Findings

Polynomial invariants of degree at most 3 separate orbits over the rationals.

A generalized upper degree bound for separating invariants over non-algebraically closed fields.

Extension of known results from algebraically closed fields to more general fields.

Abstract

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating invariants for representations of finite abelian groups over algebraically closed base fields of non-modular characteristic is generalized for the case of base fields that are not algebraically closed (like the fields of real or rational numbers).