General degree formulas for tree tensor network varieties

Determine the degree of general tree tensor network varieties, including tensor-train and subspace varieties, in terms of their defining dimension and rank data, and derive a general explicit formula where possible.

Background

The degree is a fundamental projective invariant, but the paper only provides a combinatorial procedure for computing it in the special case of order-3 tensor-train varieties. The procedure uses minimal-prime descriptions of initial ideals and non-intersecting path families.

Beyond this special case, the authors state that the degree of tree tensor network varieties is not known in general. They further note that even tensor-train and subspace varieties lack a general explicit formula expressed directly in terms of their input data, although nearly explicit formulas are available in some related settings.

References

However, the degree of tree tensor network varieties is, in general, not known at this moment. Even for the special cases of tensor train and subspace varieties there is no general explicit formula that expresses the degree in terms of the data, for instance, in terms of $\mathbf{d}$ and $\mathbf{r}$ in the case of tensor train varieties.

— Equations of Tree Tensor Network Varieties  (2608.19071 - Hoşten et al., 19 Aug 2026) in Section “Degree of an order $3$ tensor train variety,” opening paragraph