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Formulae for the singularities at infinity of plane algebraic curves. (English) Zbl 1015.32026
The paper under review presents known as well as new results on invariants of plane curves at infinity, completing a list of (meanwhile classical) formulas of F. Pham [Astérisque 7/8 (1973), 363-391 (1974; Zbl 0291.14012)]. Let \(\mathbb{C}^2\to\mathbb{C}^2\) be a map given by a polynomial \(f \in C[X,Y]\) with isolated critical points. Tessier’s lemma gives a formula for \((f, \frac{\partial X}{\partial Y})\) in terms of the Milnor-number \(\mu _p(f)\) and the intersection multiplicity \((f,X-a)_p\) for \(p\in\mathbb{C}^2\) such that \(X=a\) intersects the reduced curve \(f=0\) in finitely many points.
Section 1 of the paper uses a projective version of Tessier’s lemma to obtain e.g. simple proofs of Plücker’s and Noether’s formula. After technical preparations, in section 3 a global version of Tessier’s lemma is given. The numbers \(\mu _p(f)\) and \(\lambda (f)\) (\(=\) jump of Milnor-numbers at infinity) can be described in terms of the restriction of \(f\) to the Polar \(\frac{\partial f}{\partial Y} =0\) thus giving estimates for \(\mu (f) + \lambda (f)\) and for the number of critical values at infinity.
The final sections give new proofs for several known formulas, like the description of the Euler number of the fibre \(f^{-1}(t)\). Further, there is an estimate on the Lojaciewicz exponent at infinity, followed by questions on open problems. Two appendices include a variant of the Riemann-Hurwitz formula and Newton’s method of determining orders of roots for a polynomial in a valued field.

32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
32S55 Milnor fibration; relations with knot theory
14H20 Singularities of curves, local rings
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