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Lojasiewicz numbers and singularities at infinity of polynomials of two complex variables. (Nombres de Lojasiewicz et singularités à l’infini des polynômes de deux variables complexes.) (French) Zbl 0807.32025
If \(P(x,y)\) is a polynomial of two complex variables, then, for a certain finite set \(A \subset \mathbb{C}\), the mapping \(P\): \(\mathbb{C}^ 2 - P^{-1} (A) \to \mathbb{C} - A\) is a locally trivial fibering. Points of \(A\), which are not critical values of the polynomial \(P\) in the plane \(\mathbb{C}^ 2\) are called critical values of \(P\) corresponding to singularities at infinity.
A description of critical values corresponding to singularities at infinity is given in terms of so-called Lojasiewicz numbers at infinity. Let \(t_ 0 \in \mathbb{C}\), \(\delta > 0\), \(r > 0\), \(\varphi (r)\) and \(\varphi_ \delta (r)\) be the minima of \(| \text{grad} P(x,y) |\) with respect to the sets \(\{| x,y | = r\}\) and \(\{| x,y | = r, | P(x,y) - t_ 0 | \leq \delta\}\) respectively, \({\mathcal L}_ \infty (P) = \lim_{r \to \infty} (\ln \varphi (r)/ \ln r)\), \({\mathcal L}_{\infty,t_ 0}\) \((P) = \lim_{\delta \to 0} \lim_{r \to \infty} (\ln \varphi_ \delta (r)/ \ln r)\).
The main results: The polynomial \(P\) has singularities at infinity if and only if \({\mathcal L}_ \infty (P) < -1\). A value \(t_ 0\in \mathbb{C}\) is a critical value of \(P\) corresponding to a singularity at infinity if and only if \({\mathcal L}_{\infty, t_ 0} < 0\) (in this case, automatically \({\mathcal L}_{\infty, t_ 0} < - 1)\).
An algorithm for the computation of the Lojasiewicz number \({\mathcal L}_{\infty, t_ 0}\) according to Puiseux expansions of the curves \(P(x,y) = t_ 0\) is presented.

32S05 Local complex singularities
14B05 Singularities in algebraic geometry
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)