论文标题
关于有限领域的置换有理功能的猜想
on a conjecture on permutation rational functions over finite fields
论文作者
论文摘要
令$ p $为素数，$ n $为一个积极的整数，考虑$ f_b（x）= x+（x^p-x+b）^{ - 1} \ in \ bbb f_p（x）$，其中$ b \ in \ bbb f_ {p^n} $是如此$ \ trect {p^n} $ f text tr} $众所周知，（i）$ f_b $输入$ \ bbb f_ {p^n} $ for $ p = 2,3 $和所有$ n \ ge 1 $; （ii）对于$ p> 3 $和$ n = 2 $，$ f_b $ permiputes $ \ bbb f_ {p^2} $，仅当$ \ text {tr} _ {p^2/p}（b）（b）= \ pm 1 $; （iii）对于$ p> 3 $和$ n \ ge 5 $，$ f_b $不输入$ \ bbb f_ {p^n} $。已经猜想，对于$ p> 3 $和$ n = 3,4 $，$ f_b $不输入$ \ bbb f_ {p^n} $。我们证明了这个猜想的足够大$ P $。
Let $p$ be a prime and $n$ be a positive integer, and consider $f_b(X)=X+(X^p-X+b)^{-1}\in \Bbb F_p(X)$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. It is known that (i) $f_b$ permutes $\Bbb F_{p^n}$ for $p=2,3$ and all $n\ge 1$; (ii) for $p>3$ and $n=2$, $f_b$ permutes $\Bbb F_{p^2}$ if and only if $\text{Tr}_{p^2/p}(b)=\pm 1$; and (iii) for $p>3$ and $n\ge 5$, $f_b$ does not permute $\Bbb F_{p^n}$. It has been conjectured that for $p>3$ and $n=3,4$, $f_b$ does not permute $\Bbb F_{p^n}$. We prove this conjecture for sufficiently large $p$.
