Algebras Generated by Reciprocals of Linear Forms

By Hiroaki Terao. 2001

66 citations recorded by the source corpus.

Let Δ be a finite set of nonzero linear forms in several variables with coefficients in a field K of characteristic zero. Consider the K-algebra C(Δ) of rational functions generated by {1/α ∣ α ∈ Δ}. Then the ring ∂(V) of differential operators with constant coefficients naturally acts on C(Δ). We study the graded ∂(V)-module structure of C(Δ). We especially find standard systems of minimal generators and a combinatorial formula for the Poincare series of C(Δ). Our proofs are based on a theorem by Brion–Vergne [4] and results by Orlik–Terao [9].

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e6c831c9a75752896ec903ccf581187095612b48

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

The maximum likelihood degree (2004, background): “However, for other combinations of plane curves the ML degree and the number of bounded regions diverge, and we prove a tight upper bound on the latter in Theorem 12. Also, following work of Terao [ 24 ] and Varchenko [25], we show in Theorem 13 that the ML degree coincides with the number of bounde”
Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry (2009, background): “The degree of the projective variety L−1 and its prime ideal PL are known from work of Terao (2002) and its refinements due to Proudfoot and Speyer (2006).”
A Broken Circuit Ring (2004, methodology): “The Hilbert series of R(L) has already been computed by Terao [Te], using different methods.”
Combinatorics and geometry of power ideals (2008, background): “• (Berget [4], Brion-Verge [5], Orlik-Terao [13], Proudfoot-Speyer [17], Terao [23]) Given a hyperplane arrangement determined by the linear functionals α1, . . . , αn, various subalgebras of the algebra generated by 1α1 , . . . , 1 αn have been studied, in some cases with additional structure.”