External points to a conic from a Baer subplane

TL;DR

This paper investigates the number of external points to a conic in a finite Desarguesian plane, specifically when the conic intersects a Baer subplane, providing a complete classification of possible configurations.

Contribution

It offers a complete classification of the possible counts of external points to a conic sharing at least one point with a Baer subplane in finite Desarguesian planes.

Findings

Classified all possible external point counts for intersecting conics and Baer subplanes.

Provides explicit conditions for the configurations in finite Desarguesian planes.

Advances understanding of geometric configurations in finite projective planes.

Abstract

For an irreducible conic $\mathcal C$ in a Desarguesian plane of odd square order, estimating the number of points from a Baer subplane which are external to $\mathcal C$ is a natural problem. In this paper, a complete list of possibilities is determined for the case where $\mathcal C$ shares at least one point with the subplane.