No abstract

We present a spectral sequence which efficiently computes Betti numbers of a closed semi‐algebraic subset of ℝPn defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in ℝPn. We do not restrict ourselves to the term E2 of the spectral sequence and give a simple explicit formula for the differential d2.

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 initiate the study of average intersection theory in real Grassmannians. We define the expected degree{\operatorname{edeg}G(k,n)} of the real Grassmannian {G(k,n)} as the average number of real k-planes meeting nontrivially {k(n-k)} random subspaces of {\mathbb{R}^{n}}, all of dimension {n-k}, where these subspaces are sampled uniformly and independently from {G(n-k,n)}. We express {\operatorname{edeg}G(k,n)} in terms of the volume of an invariant convex body in the tangent space to the Grassmannian, and prove that for fixed {k\geq 2} and {n\to\infty},\operatorname{edeg}G(k,n)=\deg G_{\mathbb{C}}(k,n)^{\frac{1}{2}\varepsilon_{k}% +o(1)},where {\deg G_{\mathbb{C}}(k,n)} denotes the degree of the corresponding complex Grassmannian and {\varepsilon_{k}} is monotonically decreasing with {\lim_{k\to\infty}\varepsilon_{k}=1}. In the case of the Grassmannian of lines, we prove the finer asymptotic\operatorname{edeg}G(2,n+1)=\frac{8}{3\pi^{5/2}\sqrt{n}}\biggl{(}\frac{\pi^{2}% }{4}\biggr{)}^{n}(1+\mathcal{O}(n^{-1})).The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set {X\subseteq\mathbb{R}\mathrm{P}^{n-1}} of dimension {n-k-1} its Chow hypersurface {Z(X)\subseteq G(k,n)}, consisting of the k-planes A in {\mathbb{R}^{n}} whose projectivization intersects X. Denoting {N:=k(n-k)}, we show that\mathbb{E}\#(g_{1}Z(X_{1})\cap\cdots\cap g_{N}Z(X_{N}))=\operatorname{edeg}G(k% ,n)\cdot\prod_{i=1}^{N}\frac{|X_{i}|}{|\mathbb{R}\mathrm{P}^{m}|},where each {X_{i}} is of dimension {m=n-k-1}, the expectation is taken with respect to independent uniformly distributed {g_{1},\dots,g_{m}\in O(n)} and {|X_{i}|} denotes the m-dimensional volume of {X_{i}}.