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Une formule de Jacobi et ses conséquences. (A Jacobi formula and its consequences). (French) Zbl 0742.32004
The authors generalize the classical Jacobi formula for a polynomial mapping \(P=(P_ 1,\dots,P_ n): \mathbb{C}^ n\to\mathbb{C}^ n\) and a polynomial \(Q\), concerning the sum of residues of the meromorphic \(n\)- form \(\omega=(Q/P_ 1\cdots P_ n)dz_ 1\land\cdots\land dz_ n\), to the case when \(P\) may have zeros at infinity. Namely: if \(\deg(P_ i)\leq D\), \(i=1,\dots,n\), and there exist strictly positive numbers \(K\), \(\kappa\), \(d\) such that \[ \| P(z)\|\geq\kappa\| z\|^ d\quad\hbox{for}\quad \| z\|\geq K \] (this is equivalent to the fact that the mapping \(P\) is proper), then, for any \(k_ 1,\dots,k_ n\in\mathbb{N}\cup\{0\}\) such that \((k_ 1+\cdots+k_ n+n)d>\deg(Q)+(n-1)(D- d)+n\), the Jacobi formula \[ \sum_{A\in P^{-1}(0)}[\hbox{res}_ A(Q/P_ 1^{k_ 1+1}\cdots P_ n^{k_ n+1})]dz_ 1\land\cdots\land dz_ n=0 \] holds.
Applications of this formula to the Nullstellensatz in \(\mathbb{C}[X_ 1,\dots,X_ n]\) are given, too.

MSC:
32A27 Residues for several complex variables
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References:
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