2025/02/28 by Eur, Christopher, Lam, Thomas · 1 citation
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.20782
Positive geometries are semialgebraic sets equipped with a canonical differential form whose residues mirror the boundary structure of the geometry. Every full-dimensional projective polytope is a positive geometry. Motivated by the canonical forms of polytopes, we construct a canonical form for any tope of an oriented matroid, inside the Orlik--Solomon algebra of the underlying matroid. Using these canonical forms, we construct bases for the Orlik--Solomon algebra of a matroid, and for the Aomoto cohomology. These bases of canonical forms are a foundational input in the theory of matroid amplitudes introduced by the second author.