On the distribution of polynomials with bounded height

TL;DR

This paper derives asymptotic formulas for the probability that a random polynomial with bounded integer coefficients has a specific signature, extending previous theoretical results in polynomial distribution analysis.

Contribution

It provides new asymptotic expressions and formulas for the distribution of polynomial signatures with bounded coefficients, building on and extending prior research.

Findings

Asymptotic probability formulas for polynomial signatures

Extensions of earlier theoretical results by Akiyama, Peth\

Numerical results supporting the formulas

Abstract

We provide an asymptotic expression for the probability that a randomly chosen polynomial with given degree, having integral coefficients bounded by some B, has a prescribed signature. We also give certain related formulas and numerical results along this line. Our theorems are closely related to earlier results of Akiyama and Peth\H{o}, and also yield extensions of recent results of Dubickas and Sha.