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 ${\mathcal {T}}$ of a square must satisfy a single irreducible homogeneous polynomial relation $p({\mathcal {T}})$ depending only on the combinatorics of ${\mathcal {T}}$. The invariant $p({\mathcal {T}})$ is called the Monsky polynomial; it captures algebraic, geometric, and combinatorial information about ${\mathcal {T}}$. We give an algorithm that computes a lower bound on the degree of $p({\mathcal {T}})$, and we present several examples in which the algorithm is used to compute the degree.