Arnold's Problems ---2001 typeset by S. V. Duzhin 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 xy . 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)) Ÿ (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 Collections: Mathematics