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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

Citations:

Zbl 0209.36001
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