Abstract
In his 16th problem, Hilbert asked to study the topological structure of the level lines of real polynomials of n variables. Our goal is to show that the topological classification of the real polynomials defining these real algebraic curves is a richer problem. For instance, there are 17746 smooth Morse functions on S2 having T=4 saddles (the maximum value for the fourth-degree polynomials on ℝ2). The ergodic theory of random graphs, basic for this study, suggests the growth rate T2T for a large number T of saddles, and we prove the lower and upper bounds of the orders TT and T2T for the number of topological types.
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