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Über den $$n$$-dimensionalen Residuensatz. (On the $$n$$-dimensional residuum theorem). (German) Zbl 0789.14003
The author gives a survey on geometric applications of the residue theorem in the $$n$$-dimensional affine or projective space. Let $$f:=(f_ 1,\dots,f_ n)$$ be a sequence of polynomials in the polynomial ring $$K[X_ 1,\dots,X_ n]$$ over an algebraically closed field $$K$$ such that the homogeneous components form a regular sequence $$Gf=(Gf_ 1,\dots,Gf_ n)$$. Then $$(f_ 1,\dots,f_ n)$$ is also a regular sequence, and the algebraic set $${\mathcal V} (f_ 1, \dots, f_ n) \subset \mathbb{A}^ n$$ consists of finitely many closed points $$P_ i$$. For any differential form $$\omega=hd X_ 1 \dots dX_ n$$ and each point $$P \in {\mathcal V} (f_ 1,\dots,f_ n)$$ there is the Grothendieck residue $$\text{Res}_ P [{\omega \over f}]$$, for which the author cites several sources in the literature, especially the elementary construction of Scheja and Storch. (In the classical case $$K=\mathbb{C}$$ they coincide with the analytically defined residues.) One defines $$G \omega:=Gh \cdot dX_ 1 \dots dX_ n$$ and $$\det \omega:=\deg h+ \sum^ n_{i= 1} \deg X_ i$$. Then the residue theorem states that for $$\deg \omega \leq \sum^ n_{i=1} \deg X_ i$$ one has $\sum_{P \in {\mathcal V} (f)} \text{Res}_ P \left[ {\omega \over f}\right]=\text{Res}_ O \left[ {G \omega \over Gf} \right].$ The right hand side of this equation vanishes for $$\deg \omega< \sum^ n_{i=1} \deg X_ i$$. If one takes the standard graduation $$\deg X_ i=1$$ for all $$i$$ then the left hand side of the residue theorem depends only on the points at infinity of the hypersurfaces $$h=0$$ und $$f_ i=0$$, $$i=1,\dots,n$$.
By choosing suitable differential forms $$\omega$$ which allow a geometric interpretation one can get many classical results in intersection theory of hypersurfaces in $$\mathbb{A}^ n$$ and generalizations of them as corollaries of the residue theorem. The author gives several impressing examples. Among others he proves a theorem of Newton, a formula of Reiss (1837), a formula of Jacobi (1835), a theorem of Humbert (1885) and even the well known theorems of Pappus and Pascal. He also gives generalizations of some of these theorems.

##### MSC:
 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14A05 Relevant commutative algebra 13N05 Modules of differentials
##### Keywords:
differential form; Grothendieck residue