Universality theorems for zeros of random real polynomials with fixed coefficients

Consider a monic polynomial of degree \(n\) whose subleading coefficients are independent, identically distributed, nondegenerate random variables having zero mean, unit variance, and finite moments of all orders, and let \(m \geq 0\) be a fixed integer. We prove that such a random monic polynomial...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: King, Matthew C, Swaminathan, Ashvin
Format: Article
Language:English
Subjects:
Independent variables
Integers
Polynomials
Random variables
Online Access:Get full text
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Summary:Consider a monic polynomial of degree \(n\) whose subleading coefficients are independent, identically distributed, nondegenerate random variables having zero mean, unit variance, and finite moments of all orders, and let \(m \geq 0\) be a fixed integer. We prove that such a random monic polynomial has exactly \(m\) real zeros with probability \(n^{-3/4+o(1)}\) as \(n\to \infty\) through integers of the same parity as \(m\). More generally, we determine conditions under which a similar asymptotic formula describes the corresponding probability for families of random real polynomials with multiple fixed coefficients. Our work extends well-known universality results of Dembo, Poonen, Shao, and Zeitouni, who considered the family of real polynomials with all coefficients random. As a number-theoretic consequence of these results, we deduce that an algebraic integer \(\alpha\) of degree \(n\) has exactly \(m\) real Galois conjugates with probability \(n^{-3/4+o(1)}\), when such \(\alpha\) are ordered by the heights of their minimal polynomials.
ISSN:2331-8422