Generalized Dissections and Monsky’s Theorem

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