For every tensor T and every tensor eigenvalue lambda, the algebraic multiplicity dominates the geometric and span multiplicities: operatornameam_T(lambda)≥ operatornamegm_T(lambda) and operatornameam_T(lambda)≥ operatornamesm_T(lambda), equivalently operatornameam_T(lambda)≥ maxoperatornamegm_T(lambda),operatornamesm_T(lambda).

Conjectures on algebraic, geometric, and span multiplicities of tensor eigenvalues

For every tensor $T$ and every tensor eigenvalue $\lambda$ , the algebraic multiplicity dominates the geometric and span multiplicities: $\operatorname{am}_T(\lambda)\geq \operatorname{gm}_T(\lambda)$ and $\operatorname{am}_T(\lambda)\geq \operatorname{sm}_T(\lambda)$ , equivalently $\operatorname{am}_T(\lambda)\geq \max\{\operatorname{gm}_T(\lambda),\operatorname{sm}_T(\lambda)\}$ .

References

Primary source

arXiv

Additional references

Resultant multiplicity via projective degrees and applications to tensor eigenvalues — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed preprint claims to settle the tensor eigenvalue multiplicity conjectures, but the claim has not been independently verified.

The problem collects Hu–Ye-type conjectures concerning algebraic, geometric, and span multiplicities of tensor eigenvalues. Earlier literature records proofs in several special cases, but not a general resolution.

September 2026 claimed resolution

The preprint Resultant multiplicity via projective degrees and applications to tensor eigenvalues claims a general resultant-multiplicity bound for arbitrary-dimensional projective zero schemes and transfers it to tensor eigenschemes, asserting that the tracked conjectures are settled. This claim is unverified.

Current status (as of September 2026): A preprint claims a general solution, but no independent verification is recorded, so the conjectures remain unsettled.

Sources

No solutions have been posted yet.