我們主要在研究在q個元素的有限體Fq上絕對不可分多項式F(T,x)的monic因式個數和的平均數。令p是Fq的特徵。假設F(T,x)=f(x)是一個x為變數，Fq[T]為係數的r(1<=r<p)次多項式。在一般的情況，我們得到：
存在兩個常數c_1和c_2(和q,r和max{deg a_i}有關)使得
c_1Nq^N<= sump_{deg a=N} au(f(a))<= c_2Nq^N,
其中 au(f(a))代表f(a)的monic因式個數，而這裏的sump是跑所有的monic多項式。在二次的情況，我們令q是奇數，那麼我們就得到：
sump_{deg a=N} au(a_2a^2+a_1a+a_0)=CNq^N+O(q^N),
其中大O的常數與q和max{deg a_i}有關，而常數C我們可以完全把它寫出來。

We study the average number of monic polynomial divisors of an absolutely irreducible polynomial F(T,x) over the finite field Fq with q elements. Let p be the characteristic of F_q.
Suppose F(T,x)=f(x) is of degree r (1<= r<p) in x over F_q[T].
We show that, in general case, there exist two positive constants c_1 and c_2 (depending on q, r and max{deg a_i}) such that
c_1Nq^N<= sump_{deg a=N} au(f(a))<= c_2Nq^N,
where au(f(a)) denotes the number of monic polynomial divisors of f(a) and the sum runs through all monic polynomials in F_q[T]. In quadratic case, if q is odd, then we have
sump_{deg a=N} au(a_2a^2+a_1a+a_0)=CNq^N+O(q^N),
where the implied constant depends on q and max{deg a_i} and the constant C can be determined explicitly.

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