Spaces of polygonal triangulations and Monsky polynomials

TL;DR

This paper explores the geometric and algebraic properties of polygonal triangulations, introducing Monsky polynomials that encode combinatorial and geometric information, and provides algorithms to compute their degrees.

Contribution

It defines Monsky polynomials for generalized triangulations and offers an algorithm to estimate their degrees, linking combinatorics, geometry, and algebra.

Findings

Monsky polynomial captures key properties of triangulations

Algorithm provides lower bounds on polynomial degrees

Examples demonstrate the computation of degrees

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.