We study the statistics of the number of connected components and the volume of a random real algebraic hypersurface in RP n defined by a homogeneous polynomial f of degree d in the real Fubini-Study ensemble. We prove that for the expectation of the number of connected components:(1)Eb0(Z RP n (f )) = Θ(d n ), the asymptotic being in d for n fixed. We do not restrict ourselves to the random homogeneous case and we consider more generally random polynomials belonging to a window of eigenspaces of the Laplacian on the sphere S n , proving that the same asymptotic holds. As for the volume properties we prove that:(2) EVol(Z RP n (f )) = Θ(d). Both equations (1) and (2) exhibit expectation of maximal order in light of Milnor's bound b0(Z RP n (f )) ≤ O(d n ) and the bound Vol(Z RP n (f )) ≤ O(d).

We demonstrate counterexamples to Wilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic field is bounded then it must have codimension at least two. Examples are provided to show that this conclusion cannot be improved.

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 $d \rightarrow \infty$), 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 $n \rightarrow \infty$).Comment: 24 pages, 1 figure. Now published in the Journal of Geometry and Physic

Abstract. We investigate a problem posed by L. Hauswirth, F. Hélein, and F. Pacard [19], namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. As they suggested, we show, under some a priori assumptions, that there are only three exceptional domains: the exterior of a disk, a halfplane, and a nontrivial example found in [19] that is the image of the strip | ζ| < π/2 under ζ → ζ + sinh(ζ). We show that in R 4 this example does not have any axially symmetric analog containing its own axis of symmetry.

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