Holonomy Chen ranks conjecture for simple matroids

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Let M\mathsf{M} be a simple matroid. For each sub-matroid M′⊂M\mathsf{M}'\subset \mathsf{M}, let nk(M′)n_k(\mathsf{M}') be the number of kk-multinets supported by M′\mathsf{M}', and let θr(M)\theta_r(\mathsf{M}) denote the holonomy Chen ranks. Holonomy Chen ranks conjecture. For r≫0r\gg 0,

θr(M)=(r−1)∑M′⊂M∑k≥3nk(M′)(k+r−3r).\theta_r(\mathsf{M})=(r-1)\sum_{\mathsf{M}'\subset \mathsf{M}}\sum_{k\ge 3}n_k(\mathsf{M}')\binom{k+r-3}{r}.

This generalizes the Chen Ranks conjecture for arrangement groups. The supplied text does not state whether the conjecture has been resolved.

References

Primary source

Alexandru Suciu, “Resonance varieties and Lie algebras of matroids and hyperplane arrangements”, arXiv:2509.24060 (2025).

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