Holonomy Chen ranks conjecture for simple matroids

Let M\mathsf{M} be a simple matroid. For each sub-matroid MM\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 r0r\gg 0,

θr(M)=(r1)MMk3nk(M)(k+r3r).\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.

Sources & referencesView supporting material

Primary source

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

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