论文标题
关于有限字段的迭代四边形的分解的注释
A note on the factorization of iterated quadratics over finite fields
论文作者
论文摘要
在有限特征的有限字段上，让$ f $为一元二次多项式。 2012年，波士顿和琼斯根据$ f $的临界后轨道构建了马尔可夫流程，并推测其限制分配解释了大型迭代元素的分解$ f $。后来，夏，波士顿和作者进行了广泛的岩浆计算，发现了一些典型的四边形家族，这些家族似乎并未遵循波士顿和琼斯猜想的原始马尔可夫模型。他们通过经验观察到波士顿 - 琼斯模型预测的某些分解模式似乎从未发生过这些多项式的某些分解模式，并提出了一个多步马尔可夫模型，从而考虑了这些缺失的分解模式。在本说明中，我们为所有这些缺失的分解模式提供了证明。这些是第一个可证明的结果，可以解释为什么波士顿和琼斯的原始猜想并不适用于所有一元二次多项式。
Let $f$ be a monic quadratic polynomial over a finite field of odd characteristic. In 2012, Boston and Jones constructed a Markov process based on the post-critical orbit of $f$, and conjectured that its limiting distribution explains the factorization of large iterates of $f$. Later on, Xia, Boston, and the author did extensive Magma computations and found some exceptional families of quadratics that do not seem to follow the original Markov model conjectured by Boston and Jones. They did this by empirically observing that certain factorization patterns predicted by the Boston-Jones model never seem to occur for these polynomials, and suggested a multi-step Markov model which takes these missing factorization patterns into account. In this note, we provide proofs for all these missing factorization patterns. These are the first provable results that explain why the original conjecture of Boston and Jones does not hold for all monic quadratic polynomials.
