Journal of Commutative Algebra

A Gröbner basis for the graph of the reciprocal plane

Alex Fink, David E. Speyer, and Alexander Woo

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Given the complement of a hyperplane arrangement, let Γ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of  Γ in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.

Article information

J. Commut. Algebra, Volume 12, Number 1 (2020), 77-86.

Received: 13 May 2017
Revised: 11 June 2017
Accepted: 18 June 2017
First available in Project Euclid: 13 May 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 05E45: Combinatorial aspects of simplicial complexes 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]

characteristic polynomial reciprocal plane hyperplane arrangement no broken circuit complex


Fink, Alex; Speyer, David E.; Woo, Alexander. A Gröbner basis for the graph of the reciprocal plane. J. Commut. Algebra 12 (2020), no. 1, 77--86. doi:10.1216/jca.2020.12.77.

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