The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915187933577216 |
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| author | Yoshizaki, Hyuga |
| author_facet | Yoshizaki, Hyuga |
| contents | Let $p$ be a prime number. The $p$-power cyclic resultant of a polynomial is the determinant of the Sylvester matrix of $t^{p^n}-1$ and the polynomial. It is known that the sequence of $p$-power cyclic resultants and its non-$p$-parts converge in $\mathbb{Z}_p$. This article shows the $p$-adic convergence of the iterated $p$-power cyclic resultants of multivariable polynomials. As an application, we show the $p$-adic convergence of the torsion numbers of $\mathbb{Z}_p^d$-coverings of links. We also explicitly calculate the $p$-adic limits for the twisted Whitehead links as concrete examples. Moreover, in a specific case, we show that our $p$-adic limit of torsion numbers coincides with the $p$-adic torsion, which is a homotopy invariant of a CW-complex introduced by S. Kionke. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06194 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials Yoshizaki, Hyuga Number Theory 13P15, 11C08, 11R09, 57K10, 57K31 Let $p$ be a prime number. The $p$-power cyclic resultant of a polynomial is the determinant of the Sylvester matrix of $t^{p^n}-1$ and the polynomial. It is known that the sequence of $p$-power cyclic resultants and its non-$p$-parts converge in $\mathbb{Z}_p$. This article shows the $p$-adic convergence of the iterated $p$-power cyclic resultants of multivariable polynomials. As an application, we show the $p$-adic convergence of the torsion numbers of $\mathbb{Z}_p^d$-coverings of links. We also explicitly calculate the $p$-adic limits for the twisted Whitehead links as concrete examples. Moreover, in a specific case, we show that our $p$-adic limit of torsion numbers coincides with the $p$-adic torsion, which is a homotopy invariant of a CW-complex introduced by S. Kionke. |
| title | The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials |
| topic | Number Theory 13P15, 11C08, 11R09, 57K10, 57K31 |
| url | https://arxiv.org/abs/2503.06194 |