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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
##### Keywords:
residue; Jacobi formula; Nullstellensatz
Full Text:
##### References:
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