ИСТИНА

The talk is devoted to a problem related to Hilbert's 16th oval problem. We show that any arrangement of ovals in the plane can be realized (up to isotopy) as an algebraic curve of degree 2r, where r is the number of ovals. Moreover, there exists a realizing polynomial of the form |P|^2-|Q|^2, for some coprime polynomials P = P(z), Q = Q(z) of degrees r = deg P > deg Q, whose roots together form an r-point configuration in the plane. Moreover, the degree 2r of the curve is the best for polynomials of this form |P|^2-|Q|^2, i.e., for any arrangement of ovals, it cannot be reduced while preserving this form of the realizing polynomial. Thus, we get a 2r-parameter family of algebraic functions F=|P/Q|^2 (parametrized by r-point configuration in the plane) realizing the given arrangement of r ovals as its level line {F=const}. Almost all of these functions are Morse. All these Morse functions are minimal in the sense that they have the minimal number of critical points (equal to 2r) over all Morse functions on the 2-sphere realizing the given arrangement of ovals. We prove that this 2r-parameter family has the same topology as the space of all minimal Morse functions on the 2-sphere realizing the given arrangement of ovals, equipped with C∞-topology. We also give a positive answer to a question by V.I.Arnold about realizability of Morse functions on the 2-sphere by algebraic functions. Moreover, we extend this to all smooth functions (not necessarily Morse). Namely, we prove that any smooth function F with k critical points on the 2-sphere is fibrewise equivalent to a function |P/Q|^2, where max{2, k-1} > deg P > deg Q and all critical values of P/Q are real and non-negative. If F has exactly 3 critical values (thus, F corresponds to a dessin d'enfant on S^2), then our function P/Q is a Belyi map (i.e., a non-constant holomorphic map of the Riemann sphere to itself that is unramified away from 0, 1 and the infinity) corresponding to this dessin d'enfant.