Iteration of Quadratic Polynomials Over Finite Fields
D. R. Heath-Brown
备注：
This revision acknowledges prior work by Juul, Kurlberg, Madhu and Tucker, and by Shao, proving results that are closely related to those of the current paper
对于奇数基数的有限域$q$，我们证明了$q$的迭代序列，从$0$开始，总是在$O(q/\log\log q)$步之后递归.对于$X^2+1$，任何起始值都是如此。当$q$为2 mod 3时，传统的“生日paradise”模型不适用于$X^3+c$的迭代.
英文摘要：
For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX^2+c$, starting at $0$, always recurs after $O(q/\log\log q)$ steps. For $X^2+1$ the same is true for any starting value. We suggest that the traditional "Birthday Paradox" model is inappropriate for iterates of $X^3+c$, when $q$ is 2 mod 3.
We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group generated by shift and inversion maps is used to study the randomness properties of permutation polynomials.
We estimate the number of primes represented by a general quadratic polynomial with discriminant $Δ$, assuming that the corresponding real character is exceptional.
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number
我们考虑定义在由x --> x * ceil（x）给出的有理数上的映射X，其中ceil（x）表示大于或等于x的最小整数，并研究对于每个有理数，找到最终将其发送到整数的X的最小迭代次数的问题。给定两个自然数M和n，我们证明了分母为M的不可约分数的集合，其轨道通过X恰好在n次迭代中达到整数，是模M^n的同余类的不交并，并建立了确定它们的有限过程。我们还描述了一个有效的算法，以决定如果一个轨道未能击中一个整数，直到一个规定的迭代次数，并推导出的概率，轨道进入一组整数等于一。
In this paper, we define the constant $D(φ, p)$, an analogue for the Davenport constant, for sequences on the finite field $\mathbb{F}_p$, defined via quadratic symmetric polynomials. Next, we state a series of results presenting either the exact value of $D(φ, p)$, or lower and upper bounds for this constant.
We provide a number of conditions on the rational numbers $u$ and $v$ which ensure that the Laurent series $g_{u,v}(x):=\prod_{t=0}^\infty (1+ux^{-3^t} + vx^{-2\cdot 3^t})$ is badly approximable.
We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.
