On the number of connected components of random algebraic hypersurfaces

Abstract

We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}{P}^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. The classification due to E. Kostlan shows that specifying an invariant ensemble is equivalent to assigning a weight to each eigenspace of the spherical Laplacian. Fixing $n$, we consider a family of invariant ensembles (choice of eigenspace weights) depending on the degree $d$. Under a rescaling assumption on the eigenspace weights (as $d\to \infty$), we prove that the order of growth of $\mathbb{E}{b}_0(X)$ satisfies: $$\mathbb{E}{b}_0(X)=\Theta \left({\left[\mathbb{E}{b}_0(X\cap \mathbb{R}{P}^1)\right]}^n\right).$$ This relates the average number of components of $X$ to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial $f{|}_{\mathbb{R}{P}^1}.$ The proof requires an upper bound for $\mathbb{E}{b}_0(X)$, which we obtain by counting extrema using Random Matrix Theory methods from recent work of the first author, and it also requires a lower bound, which we obtain by a modification of the barrier method. We also provide a quantitative upper bound for the implied constant in the above asymptotic; for the real Fubini-Study model these estimates reveal super-exponential decay of the leading coefficient (in $d$) of $\mathbb{E}{b}_0(X)$ (as $n\to \infty$).

arXiv comment24 pages, 1 figure. Now published in the Journal of Geometry and Physics

References in corpus (9)

• Complexity of random smooth functions on the high-dimensional sphere
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• High-Dimensional Random Fields and Random Matrix Theory
• Systems of Quadratic Inequalities
• Statistics on Hilbert's Sixteenth Problem
• Asymptotiques de nombres de Betti d'hypersurfaces projectives réelles
• Random Matrices and complexity of Spin Glasses
• Gap probabilities and applications to geometry and random topology

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