论文标题
二项式在有限场上的稳定性
Stability of binomials over finite fields
论文作者
论文摘要
如果$ k $上的多项式$ f（x）$如果其所有迭代都不可值得$ k $，则据说是稳定的。 L. Danielson和B. Fein表明，如果$ f（x）$是不可约的元二项式，那么在$ f（x）$的情况下，它在$ k $上是稳定的。在本文中，证明这一结果不再存在于有限领域。为了使给定的二项式在$ \ mathbb {f} _q $上稳定，给出了必要和足够的条件。这些条件用于构造一张表，列出了$ \ mathbb {f} _q $的稳定二项式，$ f（x）= x^d-a $，$ a \ in \ mathbb {f} _q \ setMinus \ setminus \ {0,1 \} $，for $ q \ q \ leq 27 $ d $ d $ d $ d $ d $该论文以与Mersenne Primes的简短链接结尾。
A polynomial $f(x)$ over a field $K$ is said to be stable if all its iterates are irreducible over $K$. L. Danielson and B. Fein have shown that over a large class of fields $K$, if $f(x)$ is an irreducible monic binomial, then it is stable over $K$. In this paper it is proved that this result no longer holds over finite fields. Necessary and sufficient conditions are given in order that a given binomial is stable over $\mathbb{F}_q$. These conditions are used to construct a table listing the stable binomials over $\mathbb{F}_q$ of the form $f(x)=x^d-a$, $a\in\mathbb{F}_q\setminus\{0,1\}$, for $q \leq 27$ and $d \leq 10$. The paper ends with a brief link with Mersenne primes.
