Geometry of PCF parameters in spaces of quadratic polynomials
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918141712400384 |
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| author | DeMarco, Laura Mavraki, Niki Myrto |
| author_facet | DeMarco, Laura Mavraki, Niki Myrto |
| contents | We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z) = z^2 + c$. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in $\mathbb{C}^2$ contains infinitely many PCF pairs $(c_1, c_2)$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of $\mathbb{C}^n$ for any $n\geq 2$. Consequently, we obtain uniform bounds on the number of PCF pairs on non-special curves in $\mathbb{C}^2$ and the number of PCF parameters in real algebraic curves in $\mathbb{C}$, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_05274 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometry of PCF parameters in spaces of quadratic polynomials DeMarco, Laura Mavraki, Niki Myrto Dynamical Systems Number Theory We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z) = z^2 + c$. Ghioca, Krieger, Nguyen and Ye proved that an algebraic curve in $\mathbb{C}^2$ contains infinitely many PCF pairs $(c_1, c_2)$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of $\mathbb{C}^n$ for any $n\geq 2$. Consequently, we obtain uniform bounds on the number of PCF pairs on non-special curves in $\mathbb{C}^2$ and the number of PCF parameters in real algebraic curves in $\mathbb{C}$, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$. |
| title | Geometry of PCF parameters in spaces of quadratic polynomials |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2310.05274 |