Families of polynomials of every degree with no rational preperiodic points
Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb{Q}]$. However, the only examples of parametric families of polynomials with no preper...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2021-03-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.173/ |
| _version_ | 1797651556838932480 |
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| author | Sadek, Mohammad |
| author_facet | Sadek, Mohammad |
| author_sort | Sadek, Mohammad |
| collection | DOAJ |
| description | Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb{Q}]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb{Q}$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no rational preperiodic points for any $t\in K$. |
| first_indexed | 2024-03-11T16:17:31Z |
| format | Article |
| id | doaj.art-a89b8fbc205c451bb5164f47108298be |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:17:31Z |
| publishDate | 2021-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-a89b8fbc205c451bb5164f47108298be2023-10-24T14:18:40ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692021-03-01359219519710.5802/crmath.17310.5802/crmath.173Families of polynomials of every degree with no rational preperiodic pointsSadek, Mohammad0Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, İstanbul, 34956 TurkeyLet $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb{Q}]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb{Q}$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no rational preperiodic points for any $t\in K$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.173/ |
| spellingShingle | Sadek, Mohammad Families of polynomials of every degree with no rational preperiodic points Comptes Rendus. Mathématique |
| title | Families of polynomials of every degree with no rational preperiodic points |
| title_full | Families of polynomials of every degree with no rational preperiodic points |
| title_fullStr | Families of polynomials of every degree with no rational preperiodic points |
| title_full_unstemmed | Families of polynomials of every degree with no rational preperiodic points |
| title_short | Families of polynomials of every degree with no rational preperiodic points |
| title_sort | families of polynomials of every degree with no rational preperiodic points |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.173/ |
| work_keys_str_mv | AT sadekmohammad familiesofpolynomialsofeverydegreewithnorationalpreperiodicpoints |