Algebraic and Topological Combinatorics

Special Session at the Fall Eastern Section Meeting
of the American Mathematical Society

Williams College, Williamstown, MA
October 13-14, 2001

Hiroaki Terao
(Tokyo Metropolitan University)

Algebras generated by reciprocals of linear forms

Abstract [ps] [pdf]: Let $\Delta$ be a finite set of nonzero linear forms in several variables with coefficients in a field $\mathbf K$ of characteristic zero. Consider the $\mathbf K$-algebra $C(\Delta)$ of rational functions generated by $\{1/\alpha \mid \alpha \in \Delta \}$. Then the ring $\partial(V)$ of differential operators with constant coefficients naturally acts on $C(\Delta)$. We study the graded $\partial(V)$-module structure of $C(\Delta)$. We especially find standard systems of minimal generators and a combinatorial formula for the Poincar\'e series of $C(\Delta)$. The results are stated in terms of the arrangement $\mathcal{A}_\Delta$ associated with $\Delta$. Our proofs are based on a theorem by Brion-Vergne and results by Orlik-Terao.

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last updated: August 22, 2001