Spaces of Polygonal Triangulations and Monsky Polynomials

Given a combinatorial triangulation of an n -gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a generalized triangulation 𝒯 of a square must satisfy a single irreducible homogeneous polynomial relation p(𝒯) depending only on the combinatorics of 𝒯 . The invariant p(𝒯) is called the Monsky polynomial ; it captures algebraic, geometric, and combinatorial information about 𝒯 . We give an algorithm that computes a lower bound on the degree of p(𝒯) , and we present several examples in which the algorithm is used to compute the degree.