Upper Bounds on Polynomial Root Separation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912577176469504 |
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| author | Knapp, Greg Yip, Chi Hoi |
| author_facet | Knapp, Greg Yip, Chi Hoi |
| contents | In this paper, we consider the relationship between the Mahler measure of a polynomial and its separation. In 1964, Mahler proved that if $f(x) \in \mathbb{Z}[x]$ is separable of degree $n$, then $\operatorname{sep}(f) \gg_n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relation, and it led to questions about the optimal exponent on $M(f)$ in that relation. However, there has been relatively little study concerning upper bounds on $\operatorname{sep}(f)$ in terms of $M(f)$. In this paper, we prove that if $f(x) \in \mathbb{C}[x]$ has degree $n$, then $\operatorname{sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$, for example, if it only has real roots. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper Bounds on Polynomial Root Separation Knapp, Greg Yip, Chi Hoi Number Theory 11R06 (Primary), 11H99, 12D10 (Secondary), 30C15 In this paper, we consider the relationship between the Mahler measure of a polynomial and its separation. In 1964, Mahler proved that if $f(x) \in \mathbb{Z}[x]$ is separable of degree $n$, then $\operatorname{sep}(f) \gg_n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relation, and it led to questions about the optimal exponent on $M(f)$ in that relation. However, there has been relatively little study concerning upper bounds on $\operatorname{sep}(f)$ in terms of $M(f)$. In this paper, we prove that if $f(x) \in \mathbb{C}[x]$ has degree $n$, then $\operatorname{sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$, for example, if it only has real roots. |
| title | Upper Bounds on Polynomial Root Separation |
| topic | Number Theory 11R06 (Primary), 11H99, 12D10 (Secondary), 30C15 |
| url | https://arxiv.org/abs/2410.01126 |