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Let \(f\colon \mathbb{K}^n \to \mathbb{K} \) be a polynomial
(\(\mathbb{K}=\mathbb{R}\) or \(\mathbb{K}=\mathbb{C}\)). Over forty
years ago R. Thom proved that \(f\) is a \(C^\infty-\)fibration
outside a finite subset of the target, the smallest such a set is
called the bifurcation set of \(f\), we denote it by
\(B(f)\). In a natural way appears a fundamental question: how to
determine the set \(B(f)\).
Let us recall that in general the set \(B(f)\) is bigger than
\(K_0(f)\) - the set of critical values of \(f\). It contains also
the set \(B_\infty (f)\) of bifurcations points at infinity. Briefly
speaking the set \(B_\infty (f)\) consists of points at which \(f\)
is not a locally trivial fibration at infinity (i.e., outside a
large ball). To control the set \(B_\infty (f)\) one can use the
set of asymptotic critical values of \(f\)
\[
K_\infty (f)=\{ y \in \mathbb{K} :\exists { x_{\nu} \in
\mathbb{K}^n, $\,$ x_{\nu}\rightarrow\infty} \ s.t. \
f(x_{\nu})\rightarrow y \ and \ \Vert x_{\nu}\Vert \Vert d
f(x_{\nu})\Vert\rightarrow 0\}.
\]
If \(c\notin K_\infty (f)\), then it is usual to say that \(f\)
satisfies Malgrange's condition at \(c\). It is proved, that
\(B_\infty (f)\subset K_\infty (f)\). We call \(K(f)=K_0(f)\cup
K_\infty (f)\) the set of generalized crtitical values of
\(f\). Thus we have that in general \(B(f)\subset K(f)\). In the
case \(\mathbb{K}=\mathbb{C}\) we gave in [1] an algorithm to
compute the set \(K(f)\). In the real case, that is for a given real
polynomial \(f:\mathbb{R}^n \to \mathbb{R}\) we can compute \(K
(f_\mathbb{C})\) the set of generalized critical values of
\(f_\mathbb{C}\) which stands for the complexification of \(f\).
However in general the set \(K_\infty(f)\) of asymptotic critical
values of \(f\) may be smaller than \(\mathbb{R}\cap
K_\infty(f_\mathbb{C})\).
In the lecture we propose another approach to the computation of
generalized critical values which works both in the complex and in
the real case. The main new idea is to use a finite dimensional
space of rational arcs along which we can reach all asymptotic
critical values.
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