Quantitative bounds on integrality for post-critically finite maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917360653303808 |
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| author | Padhy, Rudranarayan Rout, Sudhansu Sekhar |
| author_facet | Padhy, Rudranarayan Rout, Sudhansu Sekhar |
| contents | Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ that contain all the archimedean places. For an integer $d \ge 2$, consider the unicritical polynomial family $f_{d,c}(z) = z^d + c$. Recently, Benedetto and Ih studied the distribution of post-critically finite parameters $c$ that are $S$-integral relative to a fixed point $α\in \overline{K}$ such that $f_{d, α}$ is not post-critically finite. In this paper, we study the quantitative aspects of their result. In particular, under some additional assumptions we establish quantitative bounds on the number of $S$-integral post-critically finite parameters in the generalized Mandelbrot set $\mathcal{M}_{d, v}$ relative to a non post-critically finite parameter $α$ as $α$ varies over number fields of bounded degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16521 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative bounds on integrality for post-critically finite maps Padhy, Rudranarayan Rout, Sudhansu Sekhar Number Theory Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ that contain all the archimedean places. For an integer $d \ge 2$, consider the unicritical polynomial family $f_{d,c}(z) = z^d + c$. Recently, Benedetto and Ih studied the distribution of post-critically finite parameters $c$ that are $S$-integral relative to a fixed point $α\in \overline{K}$ such that $f_{d, α}$ is not post-critically finite. In this paper, we study the quantitative aspects of their result. In particular, under some additional assumptions we establish quantitative bounds on the number of $S$-integral post-critically finite parameters in the generalized Mandelbrot set $\mathcal{M}_{d, v}$ relative to a non post-critically finite parameter $α$ as $α$ varies over number fields of bounded degree. |
| title | Quantitative bounds on integrality for post-critically finite maps |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.16521 |