Mikhail SODIN

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✬ The first half of Hilbert’s 16th problem: bounds and possible configurations of connected components of the zero set of an algebraic polynomial in R m . We study its statistical version and find the order of growth of the typical number of components of zero sets of smooth random functions of several real variables. The primary examples are • various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree (“Riemannian case”) • smooth Gaussian functions on the Euclidean space with translation-invariant distribution. (“Euclidean case”) Major difficulty/interest: “non-locality” of the problem (no integral formulas, like Jensen or Kac-Rice). Remark: upper bounds can be obtained from counting critical points. Lower bounds are more involved. ✫ 2 ✩ ✪
✬ ✫ Nodal portrait of a gaussian spherical harmonic of degree 40 (figure by Alex Barnett) 3 ✩ ✪
Page 1: ✬ On the number of components of
Page 5 and 6: ✬ Notation: N(R; F ) the number o
Page 7 and 8: ✬ Part II: Riemannian case X a sm
Page 9 and 10: ✬ Normalization: The functions fL
Page 11 and 12: ✬ Translation-invariant local lim
Page 13 and 14: ✬ Notation: N(fL) the number of c
Page 15 and 16: ✬ Part III. Related works: T.L.Ma
Page 17 and 18: ✬ Part IV. Examples 1. Trigonomet
Page 19 and 20: ✬ 3. Kostlan ensemble: Homogeneou
Page 21 and 22: ✬ Part V: Steps in the proof of T
Page 23 and 24: ✬ Step 3: The Kac-Rice bound for

Mikhail SODIN

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