Asymptotic behaviour of families of real curves
Let \(\{X_t\}_{t \in\mathbb{R}}\) be the family of fibers of a polynomial function \(f\) on a non-compact non-singular real algebraic surface. The authors characterize the regular fibers of \(f\) which are atypical due to their asymptotic behaviour at infinity. More precisely, the authors prove the following statement. Let \(t_0\) be a regular value of \(f\). Then the value \(t_0\) is typical (i.e., the map \(f\) is a \(C^\infty\)-fibration at \(t_0)\) if and only if the Euler characteristic \(\chi(X_t)\) is constant when \(t\) varies in some neighbourhood of \(t_0\) and there is no component of \(X_t\) which vanishes at infinity as \(t\) tends to \(t_0\). This criterion is compared in the paper with the situation in the complex case. For a family \(\{X_t\}_{t\in\mathbb{C}}\) of fibers of a polynomial function \(f:\mathbb{C}^2 \to\mathbb{C}\), \textit{Hà Huy Vui} and \textit{Lê Dũng Tráng} [Acta Math. Vietnam. 9, 21-32 (1984; Zbl 0597.32005)] proved that a reduced curve \(X_{t_0}\) is typical (i.e., \(t_0\) is typical) if and only if its Euler characteristic \(\chi(X_{t_0})\) is equal to the Euler characteristic of a general fiber of \(f\). On the other hand, a regular fiber \(X_{t_0}\) of a complex polynomial function is typical if and only if there are no vanishing cycles at infinity corresponding to this fiber [see \textit{D. Siersma} and \textit{M. Tibăr}, Duke Math. J. 80, 771-783 (1995; Zbl 0871.32024)]. The authors show that in the real case the two conditions should be considered together: Neither the constancy of Euler characteristic no non-vanishing condition implies that \(X_{t_0}\) is typical.
scientific article; zbMATH DE number 1319562