## The distribution of polynomials over finite fields. II.(English)Zbl 0241.12010

Let $$r_i = f_i/g_i$$, $$i = 1,\ldots, s$$, be a set of $$s$$ non-constant, rational functions (in their lowest terms), with coefficients in $$k$$, the finite field of $$q$$ elements. An estimate of the form $$\theta q+O(q^{1/2})$$ is proved for the number of $$\alpha$$ in $$k$$ for which, for each $$i$$, $$f_i(x) - \alpha g_i(x)$$ has a prescribed type of factorization into irreducible polynomials in $$k[x]$$. Here $$\theta = \theta (r_1, \ldots, r_s)$$ is given explicitly in terms of the Galois group of $$\prod_{i=1}^s (f_i - tg_i)$$ and the implied constant depends only on $$n = \max(\deg f_i, \deg g_i)$$.
In a previous paper (part I), Acta Arith. 17, 255–271 (1970; Zbl 0209.36001), the author investigated the case $$s = 1$$. Now it is shown, by example, in the case $$s = 2$$, that the plausible statement that $$\theta(r_1,r_2) = \theta(r_1) \theta(r_2)$$ need not, in fact, hold even when $$r_1$$ and $$r_2$$ are not obviously related; difficulties arising when, for example, the splitting fields of $$f_1-tg_1$$ and $$f_2-tg_2$$ are not linearly disjoint. Algebraic conditions are also given for a function $$r_1$$ to have its values $$r_1(\alpha)$$, where $$\alpha\in k$$, included amongst those of another function $$r_2$$. Examples are given, but, again, algebraic problems would make the general classification of such functions difficult.
Reviewer: Stephen D. Cohen

### MSC:

 11T06 Polynomials over finite fields 11T55 Arithmetic theory of polynomial rings over finite fields

Zbl 0209.36001
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