Heights in finite projective space, and a problem on directed graphs

TL;DR

This paper studies the height function in finite projective spaces, providing explicit formulas for two dimensions and applying these results to find the minimum edge deletions needed to make a directed graph acyclic.

Contribution

It introduces explicit formulas and estimates for the height function in two dimensions and applies these findings to a graph theory problem involving acyclic subgraphs.

Findings

Explicit formulas for height in 2D case.

Estimates for height function values.

Application to minimum edge deletions in directed graphs.

Abstract

Let $\F_p = \Z/p\Z$. The \emph{height} of a point $\mathbf{a}=(a_1,..., a_d) \in \F_p^d$ is $h_p(\mathbf{a}) = \min \left\{\sum_{i=1}^d (ka_i \mod p) : k=1,...,p-1\right\}.$ Explicit formulas and estimates are obtained for the values of the height function in the case $d=2,$ and these results are applied to the problem of determining the minimum number of edges the must be deleted from a finite directed graph so that the resulting subgraph is acyclic.