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

Melvyn B. Nathanson·Blair D. Sullivan
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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.

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Melvyn B. Nathanson, Blair D. Sullivan. 2007-03-14. Heights in finite projective space, and a problem on directed graphs. https://arxiv.org/abs/math/0703418

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