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 S 2 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 T 2T for a large number T of saddles, and we prove the lower and upper bounds of the orders T T and T 2T for the number of topological types.
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Arnold, V.I. Smooth functions statistics. Funct. Anal. Other Math. 1, 111–118 (2006). https://doi.org/10.1007/s11853-007-0008-6
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DOI: https://doi.org/10.1007/s11853-007-0008-6