Counting polynomials with positive roots
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911160942460928 |
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| author | Yatsyna, Pavlo Żmija, Błażej |
| author_facet | Yatsyna, Pavlo Żmija, Błażej |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting polynomials with positive roots Yatsyna, Pavlo Żmija, Błażej Number Theory 11C08, 11N45, 26D15, 11R16 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. |
| title | Counting polynomials with positive roots |
| topic | Number Theory 11C08, 11N45, 26D15, 11R16 |
| url | https://arxiv.org/abs/2509.14501 |