Universality theorems for zeros of random real polynomials with fixed coefficients

TL;DR

This paper establishes universality theorems for the probability of a fixed number of real zeros in random monic polynomials with certain fixed coefficients, extending previous results to broader polynomial families.

Contribution

It generalizes universality results for real zeros of random polynomials to include families with fixed coefficients, providing asymptotic probabilities and number-theoretic implications.

Findings

Probability of exactly m real zeros is n^{-3/4+o(1)} for large n.

Conditions identified for similar asymptotics in polynomial families with fixed coefficients.

Application to algebraic integers showing distribution of real Galois conjugates.

Abstract

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.