On coordinatising planes of prime power order using finite fields

TL;DR

This paper explores how finite fields can be used to coordinatize projective planes of prime power order, revealing restrictions on the algebraic structures involved and their geometric classifications.

Contribution

It provides a detailed analysis of the restrictions on planar ternary rings when coordinatizing prime power order planes with finite fields, linking algebraic form to geometric type.

Findings

Finite fields impose specific polynomial restrictions on PTRs.

The Lenz-Barlotti type influences the form of the PTR polynomial.

Optimal coordinatization methods are identified for certain planes.

Abstract

We revisit the coordinatisation method for projective planes. First, we discuss how the behaviour of the additive and multiplicative loops can be described in terms of its action on the "vertical" line, and how this means one can coordinatise certain planes in an optimal sense. We then move to consider projective planes of prime power order only. Specifically, we consider how coordinatising planes of prime power order using finite fields as the underlying labelling set leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider the Lenz-Barlotti type of the plane being coordinatised, deriving further restrictions on the form of the PTR polynomial.