First order joint differential projective invariants

TL;DR

This paper provides a comprehensive algebraic framework for first-order joint projective invariants of point configurations in the plane, including explicit generators and the structure of the invariant field under projective transformations.

Contribution

It introduces a complete algebraic description, explicit minimal generating sets, and the structure of the invariant field for configurations of points in the plane under projective group actions.

Findings

Explicit minimal generating set for the field of invariants.

Proof of algebraic independence of generators.

Closed-form expression for the primitive relative invariant.

Abstract

We present a complete algebraic description of the field of first-order joint projective invariants for configurations of \( n \) points in the plane, under the natural diagonal action of the projective group \( PGL(3,\mathbb{R}) \). For \( n > 1 \), we construct an explicit minimal generating set for the field of absolute invariants and prove its algebraic independence. We further determine the structure of the full field of invariants as a simple algebraic extension of field of absolute invariants, generated by a single primitive relative invariant of weight~$-1$, for which we provide a closed-form expression valid for all \( n > 1 \).