Heights in finite projective space, and a problem on 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 $h_p(\mathbf{a})=\min \left\{{\sum }_{i=1}^d(k{a}_i\mathrm{mod}p):k=1,...,p-1\right\}.$ 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.

10 pages

References in corpus (1)

• Cycles in dense digraphs

Cited by in corpus (1)

• Gaps in the Spectrum of Heights of Projective Points