Minimum-dimension degree vectors near centers of nonempty determinantal boxes

Let $\boldsymbol l$, $\boldsymbol d$, and $\boldsymbol s$ be data for a scaled multihomogeneous system. The data are determinantal if they admit a determinantal formula. A determinantal degree vector is a vector $\boldsymbol m\in\mathbb Z^r$ yielding such a formula, and the associated nonempty determinantal boxes are the boxes of Corollary~

that contain determinantal vectors. **Minimum-dimension box-center conjecture.** If the data $\boldsymbol l,\boldsymbol d,\boldsymbol s$ are determinantal, then determinantal degree vectors of minimum matrix dimension lie close to the center of the nonempty boxes of Corollary~

.

This is supported by experimental results and refines the preceding heuristic about box centers. It remains open whether closeness to a box center always characterizes a minimum-dimension formula.