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The Chow form of a reciprocal linear space. (English) Zbl 07155051
Summary: A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.

MSC:
14M12 Determinantal varieties
05B35 Combinatorial aspects of matroids and geometric lattices
14M15 Grassmannians, Schubert varieties, flag manifolds
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
13C14 Cohen-Macaulay modules
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