Counting and testing dominant polynomials

TL;DR

This paper investigates the properties of dominant polynomials with integer coefficients, providing counting results, probability estimates, and algorithms for testing dominance without root-finding.

Contribution

It introduces new methods for counting and testing dominant polynomials, including probabilistic results and algorithms that do not require root calculations.

Findings

Probability of dominance tends to 1 for certain random polynomials.

Proportion of dominant quadratic polynomials converges to a specific constant.

Algorithms for dominance testing without root-finding are developed.

Abstract

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height $H$ among all quadratic integer polynomials tends to $(41+6 \log 2)/72$ as $H \to \infty$. Finally, we will design some algorithms to test whether a given polynomial with integer coefficients is dominant or not without finding the polynomial roots.