Some $p$-ranks related to a conic in $PG(2,q)$

TL;DR

This paper investigates the ranks of submatrices of the incidence matrix in a projective plane with respect to a conic, revealing their algebraic properties over the finite field.

Contribution

It determines the ranks of nine submatrices of the incidence matrix partitioned by a conic in PG(2,q) over the finite field.

Findings

Ranks of all nine submatrices are explicitly computed.

Results depend on the properties of the conic and the finite field.

Provides new insights into the algebraic structure of incidence matrices in projective planes.

Abstract

Let $\A$ be the incidence matrix of lines and points of the classical projective plane $PG(2,q)$ with $q$ odd. With respect to a conic in $PG(2,q)$, the matrix $\A$ is partitioned into 9 submatrices. The rank of each of these submatices over $\Ff_q$, the defining field of $PG(2,q)$, is determined.