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Concentration of the number of real roots of random polynomials

Ander Aguirre, Hoi H. Nguyen, Jingheng Wang

Abstract

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some extent. Our method is elementary and works well for many models of random polynomials, with gaussian or non-gaussian coefficients.

Concentration of the number of real roots of random polynomials

Abstract

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some extent. Our method is elementary and works well for many models of random polynomials, with gaussian or non-gaussian coefficients.
Paper Structure (29 sections, 49 theorems, 268 equations)

This paper contains 29 sections, 49 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Statistics for real roots of the gaussian model
  3. Universality
  4. A new direction: concentration of the statistics
  5. New results
  6. Further remarks
  7. Supporting ingredients and proof methods
  8. Large Sieve inequalities
  9. Number of roots over small intervals
  10. Stability
  11. Repulsion
  12. Geometric concentration
  13. Proof methods
  14. Orthogonal polynomials: proof of Theorem \ref{['thm:orthogonal']}
  15. Properties of orthogonal polynomials
  16. ...and 14 more sections

Key Result

Theorem 1.2

and moreover

Theorems & Definitions (85)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • Theorem 1.11
  • Theorem 2.2
  • ...and 75 more