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Counting fibres of the Hadamard product using Bergman fans

Oliver Clarke, Sean Dewar, Matteo Gallet, Georg Grasegger, Daniel Green Tripp, Ben Smith

TL;DR

The paper introduces the flip product, a tropical-intersection invariant of matroids, to compute the generic fibre cardinality of Hadamard products of linear spaces. It proves that for representable matroids with complementary ranks, the fibre count equals M ∗ N, and provides a recursive algorithm to compute flip products, grounded entirely in tropical geometry. The framework is extended to counting symmetric and periodic framework realisations via gain-graph matroids, with a Chow-ring interpretation of flip products and connections to classical matroid invariants like the beta invariant and NBC bases. Computational results and deletion-contraction schemes for gain-graph matroids illustrate practical calculations and open avenues for further combinatorial and geometric applications.

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.

Counting fibres of the Hadamard product using Bergman fans

TL;DR

The paper introduces the flip product, a tropical-intersection invariant of matroids, to compute the generic fibre cardinality of Hadamard products of linear spaces. It proves that for representable matroids with complementary ranks, the fibre count equals M ∗ N, and provides a recursive algorithm to compute flip products, grounded entirely in tropical geometry. The framework is extended to counting symmetric and periodic framework realisations via gain-graph matroids, with a Chow-ring interpretation of flip products and connections to classical matroid invariants like the beta invariant and NBC bases. Computational results and deletion-contraction schemes for gain-graph matroids illustrate practical calculations and open avenues for further combinatorial and geometric applications.

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.

Paper Structure

This paper contains 45 sections, 42 theorems, 154 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Introducing: the flip product of matroids
  3. Main results
  4. Applications: counting symmetric/periodic framework realisations
  5. In the language of Chow rings and toric varieties
  6. Structure of paper
  7. Preliminaries
  8. Matroid notation and terminology
  9. Tropical intersection product
  10. Bergman fans
  11. Generic root counts
  12. Hadamard product of matroids
  13. Flip products of matroids
  14. Hadamard products for representable matroids
  15. Proof of Hadamard product theorem
  16. ...and 30 more sections

Key Result

Theorem 1.3

Let $U, V \subset \mathbb{C}^E$ be linear subspaces and let $M(U)$ and $M(V)$ be the representable matroids defined by $U$ and $V$, respectively. Then, for any generic $\lambda \in \mathbb{C}^E$ we have where $\sim$ is the equivalence relation on $U\times V$ where $(u,v) \sim (u',v')$ if and only if $u'=tu$ and $v'=t^{-1}v$ for some $t \in \mathbb{C}\setminus \{0\}$, i.e., the pairs $(u,v)$ and $

Figures (4)

  • Figure 1: The graph $G$ formed from a $K_4$ glued to a $C_4$.
  • Figure 2: The graph $G'$ formed from the graph $G$ by moving a single edge.
  • Figure 3: A $\mathbb{Z}_4$-gain graph (left) and a 4-fold symmetric realisation of the cover (right).
  • Figure 4: Six real realisations of the graph from \ref{['fig:symgain']}.

Theorems & Definitions (111)

  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6: Clarke et al. ClarkeDewarEtAl2025
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • Definition 2.1
  • Example 2.2
  • ...and 101 more