Generalized Dissections and Monsky's Theorem

TL;DR

This paper extends Monsky's equidissection theorem to generalized dissections, introduces a deformation-invariant polynomial relation among triangle areas, and explores their algebraic and combinatorial properties.

Contribution

It generalizes dissection concepts to include oriented triangles, constructs a deformation space as an algebraic variety, and links two polynomial relations to deepen understanding of area invariants.

Findings

Deformation space of generalized dissections is an irreducible algebraic variety.

Monsky's polynomial can be chosen to be deformation-invariant.

Polynomial p is congruent modulo 2 to a power of the sum of variables.

Abstract

Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky's polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.