Summary: Arnold's Problems --- 2001
typeset by S. V. Duzhin
September 25, 2001
Problem 1. (V. I. Arnold, A. Ortiz) Betti numbers of parabolic sets.
Let f(x; y) be a real polynomial in two variables. Denote by P (f) the set of parabolic
points on the surface fz = f(x; y)g, i.e. the zero set of the Hessian H[f ] = f xx f yy \Gamma f 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 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)) (m \Gamma 1)(m \Gamma 2)=2 + 1 = 4. There
is a wellknown construction of a polynomial (uv \Gamma '', where u = 0, v = 0 are equations
of ellipses that intersect in 4 points and '' 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 ].
Problem 2. (V. I. Arnold) Caustics of periodic functions.
Let g : S 1 ! R be a smooth function and u, v two real parameters. The plane curve