Heights in Finite Projective Space, and a Problem on Directed Graphs

Let Fp = Z/pZ. The height of a point a = (a1, . . . , ad) ∈ F d p is hp(a) = min n Pd i=1(kai mod p) : k = 1, . . . , p − 1 o . 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. 1. Heights in finite projective space Let F be a field and let F ∗ = F \{0}. For d ≥ 2, we define an equivalence relation on the set of nonzero d-tuples F d\{(0, . . . , 0)} as follows: (a1, . . . , ad) ∼ (b1, . . . , bd) if there exists k ∈ F ∗ such that (b1, . . . , bd) = (ka1, . . . , kad). We denote the equivalence class of (a1, . . . , ad) by 〈a1, . . . , ad〉. The set of equivalence classes is called the (d − 1)-dimensional projective space over the field F , and denoted P(F ). We consider projective space over the finite field Fp = Z/pZ. For every x ∈ Fp, we denote by x mod p the least nonnegative integer in the congruence class x. We define the height of the point a = 〈a1, . . . , ad〉 ∈ P(Fp) by hp(a) = min {