Zero distribution of random sparse polynomials

Abstract

Abstract. We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope P as their degree grow. We consider a large class of probability distributions including the ones induced from i.i.d. random coefficients whose distribution law has bounded density with logarithmically decaying tails as well as moderate measures defined over the projectivized space of Laurent polynomials. We obtain a quantitative localized version of Bernstein-Kouchnirenko Theorem.

Equidistribution of zeros of random holomorphic sections

“…Their results were generalized later in the setting of projective manifolds with big line bundles endowed with singular Hermitian metrics whose curvature is a Kähler current in [CM1,CM2,CM3], and to the setting of line bundles over compact normal Kähler spaces in [CMM] and in Corollary 3.2. Analogous equidistribution results for non-Gaussian ensembles are proved in [DS1,Ba1,Ba3,BL].…”

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

“…Consider an ordering with (16), and suppose that S := supp(µ) is non-pluripolar. Denote by Ω be the unbounded connected component of C d S, and by S the polynomial convex hull 3…”

Pluripotential theory and convex bodies: large deviation principle

“…Following [1], given P ⊂ (R + ) d a convex body, we say that a finite measure ν with support in a compact set K is a Bernstein-Markov measure for the triple (P, K, Q) if (3.13) holds for all p n ∈ P oly(nP ). For any P there exists A = A(P ) > 0 with P oly(nP ) ⊂ P An for all n. Thus if (K, ν, Q) satisfies a weighted Bernstein-Markov property, then ν is a Bernstein-Markov measure for (P, K,Q) whereQ = AQ.…”