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Let \(M\) be an oriented n-manifold, where \(n=2,3\). For a generic
\(f\in C^\infty(M,R^n)\), there is a discrete set \(S(f)\) of
critical points consisting of cusp points if \(n=2\), or swallowtail
points if \(n=3\).
In that case, at any \(p\in S(f)\) there exists a well-oriented
coordinate system centered at \(p\), and a coordinate system
centered at \(f(p)\), such that locally \(f\) has the form
\begin{gather}
f_\pm(x,y)=(\pm x,xy+y^3)\ \ \text{if}\ n=2,\\
f_\pm(x,y,z)=(\pm xy+x^2 z+x^4,y,z)\ \ \text{if}\ n=3,
\end{gather}
so one may associate with \(p\) a sign \(I(f,p)\in \{\pm 1\}\). In
the planar case the sign of a cusp equals the local topological
degree of \(f:(M,p)\rightarrow (R^2, f(p))\). A geometric
definition of the sign associated with a swallowtail was recently
introduced by Goryunov [1].
We shall show how to compute the number of points in \(S(f)\) having
the positive/negative sign in the case where \(f\colon
R^n\rightarrow R^n\) is a polynomial mapping in terms of signatures
of quadratic forms.
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