Matroids from irreducible varieties
Let $E$ be a finite set, let $\mathbb{F}$ be a field and let $\mathbb{F}^E$ denote the vector space with a fixed set of coordinates, indexed by $E$. For each $S \subseteq E$, $\pi_S: \mathbb{F}^E \rightarrow \mathbb{F}^S$ denotes the corresponding coordinate projection.
A variety is a subset of $\mathbb{F}^E$ defined by the vanishing of a system of polynomial functions. A variety is said to be irreducible if it is not the proper union of subvarieties. Each irreducible variety $V \subseteq \mathbb{F}^E$ defines a matroid $\mathcal{M}(V)$ on ground set $E$. In particular, $S \subseteq E$ is independent in $\mathcal{M}(V)$ if $\pi_S(V)$ has dimension $|S|$, and $S \subseteq E$ is spanning in $\mathcal{M}(V)$ if $\pi_S(V)$ has the same dimension as $V$.
Representable matroids can be constructed in this way. In particular, if $A$ is an $\mathbb{F}$-matrix with column set $E$, then its rowspan is a linear subspace $L$ of $\mathbb{F}^E$. Since linear subspaces can be defined by the vanishing of linear polynomials, $L$ is a variety and it is moreover irreducible. The matroid $\mathcal{M}(L)$ is isomorphic to the column matroid of $A$.
An example from rigidity theory
Another example of matroids from irreducible varieties comes from rigidity theory. Fix an integer $n$ and consider the following $(n+1)\times (n+1)$ Cayley-Menger matrix
$\begin{pmatrix} 0 & 1 & 1 & 1 & \cdots & 1 \\ 1 & 0 & x_{12} & x_{13} & \cdots & x_{1n} \\ 1 & x_{12} & 0 & x_{23} & \cdots & x_{2n} \\ 1& x_{13} & x_{23} & 0 & \cdots & x_{3n} \\ \vdots & \vdots &\vdots &\vdots & \ddots & \vdots \\ 1 & x_{1n} & x_{2n} & x_{3n} & \cdots & 0 \end{pmatrix}$
Every minor of such a matrix is a polynomial in the variables $\{x_{ij} \vert 1 \le i < j \le n\}$, which is in natural bijection with the edge set $E_n$ of the complete graph on vertex set $\{1,\dots,n\}$. The vanishing of the $(d+3)\times (d+3)$ minors of such a matrix define what’s often called the Cayley-Menger variety of $n$ points in $d$-dimensional space, and we denote it by ${\rm CM}_n^d$. It lives in the vector space $\mathbb{C}^{E_n}$ whose coordinates are indexed by $E_n$.
The significance of the variety ${\rm CM}_n^d$ is that if $p_1,\dots,p_n \in \mathbb{R}^d$ and $y_{ij} = \|p_i-p_j\|$, then the point $(y_{ij} | 1 \le i < j \le n)$ lies in ${\rm CM}_n^d$. Its irreducible, so we can talk about the matroid $\mathcal{M}({\rm CM}_n^d)$. A subset $S \subseteq E_n$ is spanning in $\mathcal{M}({\rm CM}_n^d)$ if and only if the graph $([n],S)$ is generically rigid in $d$-dimensional space. Informally speaking, what this means is that if one were to physically construct the graph $([n],S)$ in $d$-dimensional space using rigid bars for edges that are free to move around their incident vertices, then the result would be a rigid structure assuming that the vertices are placed in a sufficiently “generic” way.
Combinatorial shadows of geometric operations
There are many situations in algebraic geometry and its applications where one creates new irreducible varieties from old ones. It can be interesting and useful to study how these operations manifest combinatorially on the varieties’ matroids. For example, let $V \subseteq \mathbb{C}^E$ be an irreducible variety and let $S \subseteq E$. The closure of $\pi_S(V) \subseteq \mathbb{C}^E$ is also an irreducible variety, and its matroid is the restriction of $\mathcal{M}(V)$ to $S$. We will now discuss a more interesting example.
Let $V,W \subseteq \mathbb{C}^E$ be irreducible Varieties. The Hadamard product of $V$ and $W$, denoted ${V \star W}$, is defined to be the closure of the following set
$\left\{(v_e\cdot w_e)_{e \in E} \ | \ v \in V \ {\rm and} \ w \in W \right\}.$
Certain irreducible varieties whose matroids are interesting for rigidity purposes appear as a Hadamard product of two linear spaces, see [1]. The $d= 2$ case ${\rm CM}_{n}^2$ of Cayley-Menger variety is one such example. This raises the question: given two linear spaces, $L_1$ and $L_2$, can the matroid of the Hadamard product $L_1 \star L_2$ be described in terms of the individual matroids $\mathcal{M}(L_1)$ and $\mathcal{M}(L_2)$? And if so, can this be generalized to allow for more than two linear spaces? For varieties other than linear spaces?
For now, all we can say is that an old technique of Edmonds for constructing matroids from submodular functions works for the Hadamard product of two linear spaces. Let $E$ be a finite set, and let $f: 2^E \rightarrow \mathbb{Z}$ be a function satisfying the following properties:
- $f(S) \ge 0$ if $S \neq \emptyset$
- $f(S) \le f(T)$ if $S \subseteq T$, and
- $f$ is submodular.
Then the set of sets $\mathcal{I}$, defined as follows, is the independent sets of a matroid [2], which we denote by $\mathcal{M}(f)$
$\mathcal{I} := \{I \subseteq E: |I’| \le f(I’) \ {\rm for \ all \ } I’ \subseteq I\}$.
If $r_1,r_2: 2^E \rightarrow \mathbb{Z}$ are rank functions of matroids $M_1$ and $M_2$, then $r_1$ and $r_2$ satisfy the necessary properties to apply Edmonds’ construction, and $\mathcal{M}(r_i) = M_i$. The sum $r_1 + r_2$ also satisfies these conditions but is no longer the rank function of a matroid. The matroid $\mathcal{M}(r_1 + r_2)$ should be familiar to readers of this this blog especially – it is the matroid union of $M_1$ and $M_2$. The function $r_1 + r_2 – 1$ also satisfies the conditions required for Edmond’s construction. We can now state how the matroid of a Hadamard product of linear spaces relates to the matroids of the individual linear spaces.
Theorem [1]: Let $L_1,L_2 \subseteq \mathbb{C}^E$ be linear spaces and let $r_i$ denote the rank function of $\mathcal{M}(L_i)$. Then $\mathcal{M}(L_1 \star L_2) = \mathcal{M}(r_1 + r_2 -1)$.
The above theorem fails if one does not require $L_1$ and $L_2$ to be linear spaces. For example, if $L_1 = L_2 = V$ is a toric variety, then $V \star V = V$ and so the formula above cannot work. It also fails if one tries to naively generalize to more than two linear spaces. More specifically, if $L_1,\dots,L_d \subseteq \mathbb{C}^E$ are all linear spaces and $r_i$ denotes the rank function of $\mathcal{M}(L_i)$, then I once conjectured that $\mathcal{M}(L_1 \star \dots \star L_d) = \mathcal{M}(r_1 + \dots + r_d – (d-1))$. This was recently proven false in [3].
References
[1] Bernstein, Daniel Irving. “Generic symmetry-forced infinitesimal rigidity: translations and rotations.” SIAM Journal on Applied Algebra and Geometry 6.2 (2022): 190-215.
[2] Jack Edmonds. Matroids, submodular functions and certain polyhedra. Combinatorial Structures and Their Applications, pages 69–87, 1970.
[3] Antolini, Dario, Sean Dewar, and Shin-ichi Tanigawa. “Dilworth truncations and Hadamard products of linear spaces.” arXiv preprint arXiv:2508.04798 (2025).