Enumerating inherited conics in André planes of odd order

Abstract

The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an André net of $PG(2,q^t)$, $t\geq 3$ to construct an André plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an André plane. Further, the number of arcs in an André plane that are inherited in this way is enumerated.