A global view of residues in the torus
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.
scientific article; zbMATH DE number 1025693
scientific article