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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.

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