An upper bound for the number of planar lattice triangulations

We prove an exponential upper bound for the number f(m,n) of all maximal triangulations of the m × n grid: f(m,n) < 2. In particular, this improves a result of S. Yu. Orevkov [1]. We consider lattice polygons P (with vertices in Z), for example the convex hull of the grid Pm,n := {0, 1, . . . , m} × {0, 1, . . . , n}. We want to estimate the number of maximal lattice triangulations of P , i.e., triangulations using all integer points P ∩Z in P . These are exactly the unimodular triangulations, in which all the triangles have integer vertices and area 1 2 . From now on we will talk only about unimodular triangulations. Denote by f(P ) the number of (unimodular) triangulations of P and by f(n,m) the number of triangulations of Pm,n. S. Yu. Orevkov’s upper bound [1] is f(m,n) ≤ 4 . Theorem 1 The number f(P ) of maximal triangulations of a lattice polygon P is bounded by f(P ) ≤ 2 ′, where |E | is the number of inner (non-boundary) edges in any unimodular triangulation of P . In particular, the number of unimodular triangulations of the grid Pm,n is bounded by f(m,n) ≤ 2 < 2. The Haystack Approach Let P be a closed, not necessarily convex lattice polygon and int(P ) its interior. Define M := ( 2 Z \Z) ∩ int(P ), the possible midpoints of the inner edges of a lattice triangulation of P .