Positive Pl\"ucker tree certificates for non-realizability

TL;DR

The paper presents a novel algebraic method using Pl"ucker relations to certify non-realizability of certain simplicial spheres and pseudo-manifolds, providing new proofs for previously unresolved cases.

Contribution

It introduces a new technique based on monomial combinations of Pl"ucker relations to prove non-realizability of complex simplicial structures.

Findings

Proved non-realizability of Zheng's balanced 2-neighborly 3-sphere.

Established non-realizability of certain centrally symmetric spheres by Novik and Zheng.

Demonstrated the method's applicability to orientable pseudo-manifolds.

Abstract

We introduce a new method for finding a non-realizability certificate of a simplicial sphere Sigma: we exhibit a monomial combination of classical 3-term Pl\"ucker relations that yields a sum of products of determinants that are known to be positive in any realization of Sigma; but their sum should vanish, contradiction. Using this technique, we prove for the first time the non-realizability of a balanced 2-neighborly 3-sphere constructed by Zheng, a family of highly neighborly centrally symmetric spheres constructed by by Novik and Zheng, and several combinatorial prismatoids introduced by Criado and Santos. The method in fact works for orientable pseudo-manifolds, not just for spheres.