Algorithmic aspects of Newman polynomials and their divisors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911627500060672 |
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| author | Idris, Musbahu Sac-Épée, Jean-Marc |
| author_facet | Idris, Musbahu Sac-Épée, Jean-Marc |
| contents | We study the problem of determining which integer polynomials divide Newman polynomials. In this vein, we first give results concerning the $8438$ known polynomials with Mahler measure less than $1.3$. We then exhibit a list of polynomials that divide no Newman polynomial. In particular, we show that a degree-10 polynomial of Mahler measure \text{approximately} 1.419404632 divides no Newman polynomial, thereby improving the best known upper bound for any universal constant $σ$, if it exists, such that every integer polynomial of Mahler measure less than $σ$ divides a Newman polynomial. Finally, letting $l(x)$ denote Lehmer's polynomial, we explicitly construct Newman polynomials divisible by $l(x)^2$ with degrees up to $150$, and show that no Newman polynomial is divisible by $l(x)^3$ up to degree $160$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11486 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algorithmic aspects of Newman polynomials and their divisors Idris, Musbahu Sac-Épée, Jean-Marc Number Theory 11K16 We study the problem of determining which integer polynomials divide Newman polynomials. In this vein, we first give results concerning the $8438$ known polynomials with Mahler measure less than $1.3$. We then exhibit a list of polynomials that divide no Newman polynomial. In particular, we show that a degree-10 polynomial of Mahler measure \text{approximately} 1.419404632 divides no Newman polynomial, thereby improving the best known upper bound for any universal constant $σ$, if it exists, such that every integer polynomial of Mahler measure less than $σ$ divides a Newman polynomial. Finally, letting $l(x)$ denote Lehmer's polynomial, we explicitly construct Newman polynomials divisible by $l(x)^2$ with degrees up to $150$, and show that no Newman polynomial is divisible by $l(x)^3$ up to degree $160$. |
| title | Algorithmic aspects of Newman polynomials and their divisors |
| topic | Number Theory 11K16 |
| url | https://arxiv.org/abs/2601.11486 |