×

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables. (English) Zbl 1095.32012

Let \(F:\mathbb C^2\to\mathbb C\) be a polynomial of degree \(d>0\). Let \(\mathbb P^2(\mathbb C)=\mathbb C^2\cup\mathbb L_\infty\) denote the complex projective plane. Let \(\mathcal L_{p,t}(F)\) denote the Łojasiewicz exponent of \(F\) at a point \((p,t)\in \mathbb L_\infty\times \mathbb C\). Let \(F^*(X,Y,Z)\) be the homogeneous form corresponding to \(F\).
The authors show how to calculate \(\mathcal L_{p,t}(F)\) in terms of the local singularities at infinity of the pencil of projective curves \(F^*(X,Y,Z)-tZ^d\), \(t\in \mathbb C\). As a consequence, if \((p,t)\) is a critical point at infinity of \(F\) then \(\mathcal L_{p,t}(F)\) can be calculated by means of the polar invariants, as in the local case [see the article Invent. Math. 40, 267–292 (1977; Zbl 0446.32002) of B. Teissier].
Let \(\mathcal L_{\infty,t}(F)=\min\{\mathcal L_{p,t}(F):p\in\mathbb L_\infty\}\). It was proved in the paper [Kodai Math. J. 26, No. 3, 317–339 (2003; Zbl 1051.32016)] of J. Chadzyński and T. Krasiński that there is a constant \(\l_\infty(F)\geq 0\) such that \(\mathcal L_{p,t}(F)=\l_\infty(F)\) for all regular values \(t\) of the mapping \(F\). In general \(\mathcal L_\infty(F)\neq\l_\infty(F)\). In this paper the authors give a description of polynomials \(F\) with \(\l_\infty(F)=0\) and compute \(\l_\infty(F)\) in the case of one branch at infinity. The main results proven in the paper improve the results obtained in the article [C. R. Acad. Sci., Paris, Sér. I 311, No. 7, 429–432 (1990; Zbl 0807.32025)] of Há Huy Vui and in the mentioned article of Chadzyński and Krasiński and show that the Łojasiewicz exponent at infinity is a purely local notion.

MSC:

32S99 Complex singularities
14R99 Affine geometry
PDFBibTeX XMLCite
Full Text: DOI