Residues and Resultants

J. Math. Sci. Univ. Tokyo
Vol. 5 (1998), No. 1, Page 119--148.

Cattani, Eduardo ; Dickenstein, Alicia ; Sturmfels, Bernd
Residues and Resultants
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Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.

Mathematics Subject Classification (1991): Primary 14M25, 32A27; Secondary 13D25, 52B20
Mathematical Reviews Number: MR1617074

Received: 1997-02-11