有限域上具有少数不动点的Dickson多项式

Dickson Polyhomials with few Fixed Points in a Finite Field Carl Pomerance

设（ｎ，ｑ￣２─１）＝１，则Ｄｉｃｋｓｏｎ多项式Ｄ＿ｍ（ｘ，１）是有限域Ｆ＿ｑ上的一个置换多项式。本文证明了：如果ｑ是一个素数的幂（ｑ≥５）。则存在正整数ｎ，（ｎ，ｑ￣２─１）＝１，ｎ＜ｃ＿１（ｌｏｇｑ）￣ｃ２使得Ｄ＿ｎ（ｘ，１）在Ｆ￣ｑ上恰有５个不动点。这里ｃ＿１，ｃ＿２是绝对常数．

If(n,q￣2─ 1)=1 , then the Dickson polynomial D_n(x ,1) is a permutation polynomial of the finite field F_q. If the characteristic of F_q is at least 5,then D_n(x ,1) has at least 5 fixed points in F_r .It follows from a result of R. Nbauer that there is always some n where the number of fixed points is exactly 5.In this note we show that there is always a relatively small n with this property, namely there is such a nu mber n less than(c_1 log q)￣c2 for some absolu te constants ci and c_2. The proof uses sieve metheds。