arXiv subjects

Antonio Lerario

Publications and source records attributed to Antonio Lerario.

63 records · Page 4Linked to original sources

arXiv2013

Random matrices and the average topology of the intersection of two quadrics

Let X_R be the zero locus in RP^n of one or two independently and Weyl distributed random real quadratic forms (this is the same as requiring that the corresponding symmetric matrices are in the Gaussian Orthogonal Ensemble). We prove that the sum of the Betti numbers of X_R behaves asymptotically as n (when n goes to infinity). The methods we use combine Random Matrix Theory, Integral geometry and spectral sequences.

math.AT↗

arXiv2013

Statistics on Hilbert's Sixteenth Problem

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 Real Bombieri-Weyl distributed homogeneous polynomial of degree d. We prove that the expectation of the number of connected components of such hypersurface has order 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, we prove its expectation is of order d. Both these behaviors exhibit expectation of maximal order in light of Milnor's bound and the a priori bound for the volume.

math.AG↗

arXiv2013

Geometry of Maslov cycles

We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated problems, from Real Algebraic Geometry to sub-Riemannian Geometry.

math.SG↗

arXiv2012

Complexity of intersections of real quadrics and topology of symmetric determinantal varieties

Let X be the base locus of a linear system W of k quadrics. Let also S be the intersection of W with the discriminant hypersurface in the space of all homogeneous polynomials of degree two. We prove a formula relating the topology of X with the one of S and its (iterated) singular points. As a corollary we prove the sharp bound b(X)\leq O(n)^{k-1}.

math.AT↗

arXiv2012

Convex pencils of real quadratic forms

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)\leq 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b_k(X)\leq 2(k+2).

math.AG↗

arXiv2011

The total Betti number of the intersection of three real quadrics

We prove that the total Betti number of the intersection X of three quadrics in RP^n is bounded by n(n+1). This bound improves the classical Barvinok's one which is at least of order three in n.

math.AT↗

arXiv2011

Plenty of Morse functions by perturbing with sums of squares

Given a smooth function f on R^n and a submanifold M, we prove that the set of diagonal quadratic forms q such that the restriction of f+q to M is Morse is a dense set (in the n-dimensional space of diagonal quadratic forms). The standard transversality argument seems not to work and we need a more refined approach.

math.DG↗

arXiv2011

A real viewpoint on the intersection of complex quadrics and its topology

We study the relation between a complex projective set C in CP^n and the set R in RP^(2n+1) defined by viewing each equation of C as a pair of real equations. Once C is presented by quadratic equations, we can apply a spectral sequence to efficiently compute the homology of R; using the fact that the (Z_2)-cohomology of R is a free H*(C)-module with two generators we can in principle reconstruct the homology of C. Explicit computations for the intersection of two complex quadrics are presented.

math.AG↗

arXiv2010

Systems of Quadratic Inequalities

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the differential d_2.

math.AG↗