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\begin{document}
\begin{center}
{\Large\bf Arnold's Problems --- 2001}\\[3mm]
{\Large typeset by S.~V.~Duzhin}\\[2mm]
{\large September 25, 2001}
\end{center}


{\large
{\bf Problem 1.} (V.~I.~Arnold, A.~Ortiz)
{\it Betti numbers of parabolic sets.}
}
\medskip

Let $f(x,y)$ be a real polynomial in two variables.
Denote by $P(f)$ the set of parabolic points on the surface
$\{z=f(x,y)\}$, i.e. the zero set of the Hessian 
$H[f]=f_{xx}f_{yy}-f_{xy}^2$. 
Determine the maximal number of compact connected components
of the set $P(f)$ for all polynomials $f$ of given degree $n$.

This problem can be viewed as a specialization of the classical
{\it oval counting problem\/} for polynomials representable in the
form of a Hessian.

The first case when the answer is unknown is $n=4$.
Then $m=\deg H[f]=\deg f =4$, and the Harnack inequality
ensures that $b_0(P(f)) \le (m-1)(m-2)/2+1 =4$. There is a well-known
construction of a polynomial ($uv-\e$, where $u=0$, $v=0$ are equations
of ellipses that intersect in 4 points and $\e$ is a small number)
for which this estimate is attained. It is not known if it can be attained 
for polynomials of the form $H[f]$.

\bigskip

{\large
{\bf Problem 2.} (V.~I.~Arnold) {\it Caustics of periodic functions.}
}
\medskip

Let $g:S^1\to\R$ be a smooth function and $u$, $v$ two real parameters.
The plane curve
$$
  C_g = \{(u,v)\mid\mbox{ function }
  G_{u,v}(\f)=g(\f)+a\cos\f + b\sin\f \mbox{ is not Morse}\}
$$
is called the {\it caustic\/} of the function $g$.
The condition that $G_{u,v}(\f)$ is not Morse means that there exists
a value $\f\in S^1$ such that $G'_{u,v}(\f)=G''_{u,v}(\f)=0$
(derivatives over $\f$).

{\it Example.} The caustic of the function $g(\f)=\cos2\f$
is the astroid $u=-4\cos^3\f$, $v=4\sin^3\f$.

For generic (Morse) functions $g$ caustics are fronts (smooth curves 
with generic singularities) that satisfy a number of specific conditions:

1. A caustic has at least 4 cusps.

2. The number of cusps is even.

3. If $P_1$, $P_2$, ..., $P_{2n}$ are cusps, then the barycentres
of the sets $P_1$, $P_3$, ..., $P_{2n-1}$
and $P_2$, $P_4$, ..., $P_{2n}$ coincide. In particular, if $n=2$,
they form a parallelogram.

4. The alternating length of a caustic (we change sign after each
cusp) is 0.

5. From every point of the plane one can draw at least two tangents to the
caustic.

6. Caustics do not have inflexion points.
\medskip

{\bf Problem.} Describe all curves that are caustics of periodic
functions, i.e. give a necessary and sufficient condition for a
front to be a caustic.

\vfill

These are two out of five problems announced on September 25, 2001 at 
Arnold's seminar.

Home page of the seminar
\verb#http://www.botik.ru/~duzhin/arnsem.html#

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