Generic rank of the edge-length-to-volume Jacobian

Determine whether, for every k-dimensional simplicial complex X on n vertices with 2\le k\le n-2, the generic rank of the Jacobian \mathcal{C}(X,p) mapping squared edge lengths to squared k-volumes equals the matching number of the bipartite incidence graph H_{X_1,X_k} between edges and k-simplices.

Background

The matrix \mathcal{C}(X,p) is the Jacobian of the CayleyMenger map from squared edge lengths of the 1-skeleton to squared volumes of the k-simplices. The authors note that the proposed rank formula for volume rigidity would imply a corresponding formula for this Jacobian when the ambient dimension is sufficiently large, specifically d=n-1.

The conjectured formula identifies the generic rank of \mathcal{C}(X,p) with a purely combinatorial quantity: the maximum matching size in the edgesimplex incidence graph. This is stated as a special case of the preceding volume-rigidity conjecture.

References

It is natural to wonder about the rank of the matrix ${\cal C}(X,)$ for a generic vector $\inR{|X_1|}$. The following is a special case of Conjecture~\ref{conj}.

— On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$  (2503.01665 - Lew et al., 3 Mar 2025) in Section 4, Discussion, immediately after Example 1