A global view of residues in the torus
type: Item
instance of: scholarly article
title: A global view of residues in the torus
zbMATH Open document ID: 0899.14024
DOI: 10.1016/S0022-4049(97)00008-X
published in: Journal of Pure and Applied Algebra
publication date: 1998-09-15T00:00:00Z
review text: The authors consider Laurent polynomials \(f_1,\dots,f_n\) having only a finite set of common zeros on the torus \(T:=(\mathbb{C}^\ast)^n\). If \(q\) is another Laurent polynomial, then the global residue of the differential form \[ \varphi_q:=\frac{q}{f_1\cdots f_n}\frac{d t_1}{t_1}\wedge\dots\wedge\frac{d t_n}{t_n} \] is defined as the sum of the usual local ones. Now, the goal of the paper is to compare this notion for general \(f_i\) (with respect to their Newton polyhedra) with the toric residue introduced and investigated by \textit{D. A. Cox} [Ark. Mat. 34, No. 1, 73-96 (1996)], and by \textit{E. Cattini}, \textit{D. Cox} and \textit{A. Dickenstein} [Compos. Math. 108, No. 1, 35-76 (1997; Zbl 0883.14029)]. The toric variety needed for this purpose is taken from a polytope \(\Delta\) such that the compactifications of the hypersurfaces \([f_i=0]\) in \(X_\Delta\) do not meet outside the torus. This method provides a new approach to the calculation of global residues on a torus. Moreover, special investigations have been done for the case that the Newton polytopes of the \(f_i\)'s are scalar multiples of a fixed polytope.
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 32A27
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1025693
zbMATH Keywords: toric varieties
zbMATH Keywords: Laurent polynomials
zbMATH Keywords: toric residue
author: Eduardo Cattani
author: Alicia Dickenstein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1025693