Counting fibres of the Hadamard product using Bergman fans

TL;DR

This paper introduces the flip product, a new matroid invariant, to compute the fiber cardinality of the Hadamard product of linear spaces using tropical geometry and Bergman fans.

Contribution

It defines the flip product, links it to the fiber count of Hadamard products, and provides algorithms for its computation, extending matroid invariants and applications in rigidity theory.

Findings

Cardinality of generic fiber equals the flip product of matroids.

Provided a recursive algorithm for computing the flip product.

Extended realisation numbers to symmetric and periodic cases.

Abstract

We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable intersection of their (flipped) Bergman fans. Our first main result is that the cardinality of a generic fibre for the Hadamard product of linear spaces is exactly the flip product of their matroids. We also provide a recursive algorithm for computing the flip product of any pair of matroids. As an application of our techniques, we extend the notion of realisation numbers from rigidity theory to rotational-symmetric and periodic realisation numbers and we provide combinatorial algorithms to compute them. Finally, we show a number of existing matroid invariants are specialisations of the flip product, including the beta invariant.