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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
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