p-adic root separation and the discriminant of integer polynomials
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| Format: | Preprint |
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2025
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| _version_ | 1866912310672490496 |
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| author | Beresnevich, Victor Dixon, Bethany |
| author_facet | Beresnevich, Victor Dixon, Bethany |
| contents | In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with discriminant divisible by a (large) power of a fixed prime. One of the consequences of our findings is the existence, for all large $Q>1$, of $Q^{2/n}$ integer irreducible polynomials $P$ of degree $n$ and height $\asymp Q$ with an almost prime power discriminant of maximal size, that is $|D(P)|\asymp Q^{2n-2}$ and $D(P)=p^kC_P$ with $C_P\in\mathbb{Z}$ satisfying $|C_P|\ll1$. The method we use generalises the techniques used in the study of the real case [Beresnevich, Bernik and Götze, 2010 and 2016] and relies on a quantitative non-divergence estimate developed by Kleinbock and Tomanov. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03851 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | p-adic root separation and the discriminant of integer polynomials Beresnevich, Victor Dixon, Bethany Number Theory Dynamical Systems In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with discriminant divisible by a (large) power of a fixed prime. One of the consequences of our findings is the existence, for all large $Q>1$, of $Q^{2/n}$ integer irreducible polynomials $P$ of degree $n$ and height $\asymp Q$ with an almost prime power discriminant of maximal size, that is $|D(P)|\asymp Q^{2n-2}$ and $D(P)=p^kC_P$ with $C_P\in\mathbb{Z}$ satisfying $|C_P|\ll1$. The method we use generalises the techniques used in the study of the real case [Beresnevich, Bernik and Götze, 2010 and 2016] and relies on a quantitative non-divergence estimate developed by Kleinbock and Tomanov. |
| title | p-adic root separation and the discriminant of integer polynomials |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2504.03851 |