Relationship between rigidity matroids and Heron-variety algebraic matroids

Determine the precise relationship between the rigidity matroids arising from Cayley–Menger varieties (defined by edge-length data with bounded rank) and the algebraic matroids M_n of the Heron varieties X_n (the Zariski closures of the realization spaces of squared face volumes of n-simplices). Specifically, ascertain whether and how the structures of these matroids (including bases, circuits, and ranks) correspond or relate, and characterize any formal connections between them.

Background

Cayley–Menger varieties parametrize edge-length data for simplices subject to rank conditions on the Cayley–Menger matrix, and their algebraic matroids are the rigidity matroids central to rigidity theory. Heron varieties X_n encode the squared volumes of all positive-dimensional faces of an n-simplex via principal minors of the Cayley–Menger matrix, and their associated algebraic matroids M_n record algebraic dependencies among these squared face volumes.

The paper develops computational tools to paper M_n (including bases and Galois/monodromy groups) and notes the conceptual proximity to rigidity theory. However, the exact structural connection between rigidity matroids and the algebraic matroids of the Heron varieties is not established and is deferred to future research.