Counting polynomials with positive roots

2025/09/18 by Pavlo Yatsyna, Yatsyna, Pavlo, Błażej Żmija +1
Mathematics · #11C08 #11N45 #11R16 #26D15 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.14501

openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates the number of monic integer polynomials of degree n whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying Maclaurin's inequalities. For cubic polynomials, we derive a much more precise asymptotic result. Furthermore, we analyse the arithmetic properties of the discriminants of these polynomials, showing that a positive proportion of cubics have square-free discriminants.

Citations

• Simultaneous Diophantine approximation to points on the Veronese curve
• On monic abelian trace-one cubic polynomials
• Discriminants of Fields Generated by Polynomials of Given Height
• On the distribution of polynomials with bounded height
• Positive density of integer polynomials with some prescribed properties
• The density of discriminants of quintic rings and fields
• Approximate formulas for some functions of prime numbers

Related

• Counting integer polynomials with several roots of maximal modulus
• Counting and testing dominant polynomials
• A construction of polynomials with squarefree discriminants
• On the number of integer polynomials with multiplicatively dependent roots
• Squarefree values of polynomial discriminants I