Counting polynomials with positive roots

Pavlo Yatsyna; Błażej Żmija

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arXiv · 2509.14501

Counting polynomials with positive roots

Pavlo Yatsyna·Błażej Żmija

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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.

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Pavlo Yatsyna, Błażej Żmija. 2025-09-18. Counting polynomials with positive roots. https://arxiv.org/abs/2509.14501

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