Sharp defect bounds for regular-matroid Jacobians

Nifa, Anass

doi:10.5281/zenodo.23548043

Preprint · v2

Sharp defect bounds for regular-matroid Jacobians

Anass NifaPublished: 7 October 2026Updated: 22 October 2026

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Abstract

Let $M$ be a finite connected regular matroid of rank $r\ge2$ on $m$ elements, and let $e$ be the exponent of its Jacobian. The defect $\delta=er-2m$ is nonnegative. It vanishes precisely when $M$ is the cycle matroid of a complete graph with constant positive edge multiplicity. If $\delta>0$ , then $r\le\delta,\qquad e+r-1\le3\delta,\qquad m\le\delta^2.$ Equality in either the rank bound or the size bound holds precisely when $M$ is the cycle matroid of the cone over the graph obtained by doubling each edge of $K_r$ . Thus bounded positive defect gives finitely many isomorphism classes. For a totally unimodular representation $D$ of $M$ of full row rank, the lattice $\mathbb Z^r$ with Gram matrix $e(DD^{\mathsf T})^{-1}$ is integral. In this lattice, the columns of $D$ of squared length two generate an orthogonal sum of root lattices of type $A$ ; their matroid restriction is a direct sum of complete-graph matroids with constant multiplicities. At positive defect these columns span a proper subspace, whereas the remaining columns span $\mathbb R^r$ . Connected regular matroids of Jacobian exponent at most five are classified. The least exponent of a connected regular matroid which is neither graphic nor cographic is six.

MSC 2020

05B35: Combinatorial aspects of matroids and geometric lattices
05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
11H56: Automorphism groups of lattices

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23548043
All versions: 10.5281/zenodo.23224641
Subjects: Regular matroid; Jacobian; exponent; integral lattice; root system; finiteness
Language: English

Cite this paper

Anass Nifa. Sharp defect bounds for regular-matroid Jacobians. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23548043

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References

As listed in the manuscript bibliography.

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DOI