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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2405.10926 |
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| _version_ | 1866929688392237056 |
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| author | Gajek-Leonard, Rylan Tomer, Uri |
| author_facet | Gajek-Leonard, Rylan Tomer, Uri |
| contents | The $p$-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime $p$. In this article, we investigate how the Newton polygon changes under polynomial composition. If $f$ and $g$ are polynomials with rational (or $p$-adic) coefficients and the Newton polygon of $g$ is pure (has only one segment), we show under some mild conditions that the Newton polygon of $f\circ g$ is the same as that of $f$, but stretched horizontally by $\operatorname{deg}(g)$. When $f=g$, this implies that all iterates of certain pure polynomials are irreducible, recovering a classical result of Robert Odoni on the irreducibility of iterated Eisenstein polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10926 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stretching Newton polygons using pure polynomials Gajek-Leonard, Rylan Tomer, Uri Number Theory Primary 11S05, Secondary 11R09, 11S82 The $p$-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime $p$. In this article, we investigate how the Newton polygon changes under polynomial composition. If $f$ and $g$ are polynomials with rational (or $p$-adic) coefficients and the Newton polygon of $g$ is pure (has only one segment), we show under some mild conditions that the Newton polygon of $f\circ g$ is the same as that of $f$, but stretched horizontally by $\operatorname{deg}(g)$. When $f=g$, this implies that all iterates of certain pure polynomials are irreducible, recovering a classical result of Robert Odoni on the irreducibility of iterated Eisenstein polynomials. |
| title | Stretching Newton polygons using pure polynomials |
| topic | Number Theory Primary 11S05, Secondary 11R09, 11S82 |
| url | https://arxiv.org/abs/2405.10926 |