A Discrete Approach to the Poincare-Miranda Theorem

The Poincaré-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincaré-Miranda Theorem that are discrete in nature that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzański that proves the Poincaré-Miranda Theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop another new proof that relies on a polytopal generalization of Sperner’s Lemma of DeLoera Peterson Su. Finally, we extend these discrete ideas to prove the existence of a zero with the boundary condition of Morales, in dimension 2.