Antonio Lerario

Systems of Quadratic Inequalities

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP 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.

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 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. Fixing n , under some rescaling assumptions on the family of ensembles (as d → ∞ ), we prove that E b 0 ( X ) has the same order of growth as [ E b 0 ( X ∩ R P 1 ) ] n . 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 | R P 1 . The proof requires an upper bound for E b 0 ( X ) , which we obtain by counting extrema using Random Matrix Theory methods from Fyodorov (2013), and it also requires a lower bound, which we obtain by a modification of the barrier method from Lerario and Lundberg (2015) and Nazarov and Sodin (2009). We also provide quantitative upper bounds on implied constants; for the real Fubini–Study model these estimates provide super-exponential decay (as n → ∞ ) of the leading coefficient (in d ) of E b 0 ( X ) .

Geodesics and horizontal-path spaces in Carnot groups

We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure. 53C17; 37J60, 58E05

Experiments on the Zeros of Harmonic Polynomials Using Certified Counting

Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach, which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha theory to certify the results. We provide new examples of harmonic polynomials having the most extreme number of zeros known so far; we also study the mean and variance of the number of zeros of random harmonic polynomials.

On the zeros of random harmonic polynomials: The truncated model

Abstract Motivated by Wilmshursts conjecture and more recent work of W. Li and A. Wei [17] , we determine asymptotics for the number of zeros of random harmonic polynomials sampled from the truncated model, recently proposed by J. Hauenstein, D. Mehta, and the authors [10] . Our results confirm (and sharpen) their ( 3 / 2 ) -powerlaw conjecture [10] that had been formulated on the basis of computer experiments; this outcome is in contrast with that of the model studied in [17] . For the truncated model we also observe a phase-transition in the complex plane for the Kac–Rice density.

On the geometry of random lemniscates

We investigate the geometry of a random rational lemniscate

Probabilistic Schubert calculus

\Gamma

Statistics on Hilbert's Sixteenth Problem

, the level set

Convex Pencils of Real Quadratic Forms

\{|r(z)|=1\}

Remarks on Wilmshurst's theorem

on the Riemann sphere of the modulus of a random rational function