On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps
On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps
复制标题
论普南关于二次映射有理前周期点的猜想
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Patrick Ingram
中科院分区：
文献类型：
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作者：
Benjamin Hutz;Patrick Ingram
本文的目的是证明Poonen、Morton和Silverman关于二次多项式迭代下有理数的周期的猜想。特别地，Poonen猜想，对于c有理数族x^2+c中的任何映射，在有理数上至多有9个周期点。对于高度为10^8的c值，我们验证了这一猜想。对于二次数域，我们证明了q-有理周期点的精确周期的上界是6。
The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.