{"title":"后临界有限单临界多项式的有限性质","authors":"Robert L. Benedetto, Su-Ion Ih","doi":"10.4310/mrl.2023.v30.n2.a1","DOIUrl":null,"url":null,"abstract":"Let $k$ be a number field with algebraic closure $\\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\\geq 2$ and $\\alpha \\in \\bar{k}$ such that the map $z\\mapsto z^d+\\alpha$ is not postcritically finite. Assuming a technical hypothesis on $\\alpha$, we prove that there are only finitely many parameters $c\\in\\bar{k}$ for which $z\\mapsto z^d+c$ is postcritically finite and for which $c$ is $S$-integral relative to $(\\alpha)$. That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF $\\bar{k}$-rational points that are $((\\alpha),S)$-integral. We conjecture that the same statement is true without the technical hypothesis.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"4 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A finiteness property of postcritically finite unicritical polynomials\",\"authors\":\"Robert L. Benedetto, Su-Ion Ih\",\"doi\":\"10.4310/mrl.2023.v30.n2.a1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $k$ be a number field with algebraic closure $\\\\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\\\\geq 2$ and $\\\\alpha \\\\in \\\\bar{k}$ such that the map $z\\\\mapsto z^d+\\\\alpha$ is not postcritically finite. Assuming a technical hypothesis on $\\\\alpha$, we prove that there are only finitely many parameters $c\\\\in\\\\bar{k}$ for which $z\\\\mapsto z^d+c$ is postcritically finite and for which $c$ is $S$-integral relative to $(\\\\alpha)$. That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF $\\\\bar{k}$-rational points that are $((\\\\alpha),S)$-integral. We conjecture that the same statement is true without the technical hypothesis.\",\"PeriodicalId\":49857,\"journal\":{\"name\":\"Mathematical Research Letters\",\"volume\":\"4 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Research Letters\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4310/mrl.2023.v30.n2.a1\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Research Letters","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/mrl.2023.v30.n2.a1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
A finiteness property of postcritically finite unicritical polynomials
Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $\alpha \in \bar{k}$ such that the map $z\mapsto z^d+\alpha$ is not postcritically finite. Assuming a technical hypothesis on $\alpha$, we prove that there are only finitely many parameters $c\in\bar{k}$ for which $z\mapsto z^d+c$ is postcritically finite and for which $c$ is $S$-integral relative to $(\alpha)$. That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF $\bar{k}$-rational points that are $((\alpha),S)$-integral. We conjecture that the same statement is true without the technical hypothesis.
期刊介绍:
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