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Concentration inequalities for the number of real zeros of Kac polynomials

Van Hao Can, Oanh Nguyen

Abstract

We study concentration inequalities for the number of real roots of the classical Kac polynomials $$f_{n} (x) = \sum_{i=0}^n ξ_i x^i$$ where $ξ_i$ are independent random variables with mean 0, variance 1, and uniformly bounded $(2+\ep_0)$-moments. We establish polynomial tail bounds, which are optimal, for the bulk of roots. For the whole real line, we establish sub-optimal tail bounds.

Concentration inequalities for the number of real zeros of Kac polynomials

Abstract

We study concentration inequalities for the number of real roots of the classical Kac polynomials where are independent random variables with mean 0, variance 1, and uniformly bounded -moments. We establish polynomial tail bounds, which are optimal, for the bulk of roots. For the whole real line, we establish sub-optimal tail bounds.
Paper Structure (16 sections, 14 theorems, 166 equations, 1 figure)

This paper contains 16 sections, 14 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related literature
  3. Difficulties and main techniques
  4. Outline
  5. Preliminaries
  6. Kac-Rice formula
  7. Local universality
  8. The bulk
  9. Limiting Gaussian process
  10. Proof of Equation \ref{['eq:Iml']} in Theorem \ref{['thm:l_ml']}
  11. Proof of Theorem \ref{['thm:lower']}, Equation \ref{['eq:Im']} of Theorem \ref{['thm:l_ml']} and Theorem \ref{['thm:l_a']}
  12. Proof of Theorem \ref{['thm:lower']}
  13. Proof of Equation \ref{['eq:Im']} in Theorem \ref{['thm:l_ml']}
  14. Proof of Theorem \ref{['thm:l_a']}
  15. Proof of Theorem \ref{['thm:upper']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Dembopersistence Assume that the $\xi_i$ are iid with all moments finite. Then there exists a positive constant $b$ such that for all $\varepsilon\in (0, 2/\pi)$ and for all $n$,

Figures (1)

  • Figure 1: Real roots in $[0, 1]$ of 100 sampled Kac polynomials of degree 1000 and standard Gaussian coefficients. The real roots concentrate in the bulk $[1-n^{-0.2}, 1]$.

Theorems & Definitions (27)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3: Limiting Gaussian Process $(Z_t)$
  • ...and 17 more