Spaces of polygonal triangulations and Monsky polynomials

Abstract

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 \emph{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.