Irreducible values of polynomials (Q655364)

The aim of the paper is to investigate the validity of Schinzel's hypothesis H for polynomial rings over pseudo algebraically closed fields (PAC fields); that is, fields \(K\) for which \(V(K)\neq \varnothing\) whenever \(V\) is a non-void \(K\)-variety. The starting point is a theorem of Pollack, which states the following. Let \(n,B\) be positive integers, \(p\) a prime such that \(p\nmid 2n\), \(q\) a power of \(p\), \(f_1(X),\dots,f_r(X)\in{\mathbb F}_q[X]\) non-associate irreducible polynomials such that \(\sum\deg(f_i)\leq B.\) Then the number of degree \(n\) monic polynomials \(g(t)\in {\mathbb F}_q[t]\) for which all \(f_i(g(t))\) are irreducible in \({\mathbb F}_q[t]\) is \(\frac{q^n}{n^r} + O_{n,B}(q^{n-\frac 12}).\) The main improvements are the following two theorems. Theorem 1: Let \(K\) be a PAC field of characteristic \(p\geq 0\). Let \(f_1,\dots,f_r\in K[X]\) be irreducible polynomials, \(\omega_1,\dots,\omega_r\) roots of \(f_1,\dots,f_r\), respectively. Let also \(n\) be a positive integer, odd if \(p=2\). Assume further that \(K(\omega_i)\) has a separable extension of degree \(n\) for all \(i=1,\dots,r\). Then there exists a Zariski dense subset \(A\) of elements \((a_1,\dots,a_n)\in K^n\) such that for all polynomials \(g(t)=t^n+a_1t^{n-1}+\dots+a_n\) with \((a_1,\dots,a_n)\in A\), all the polynomials \(f_i(g(t))\) are irreducible in \(K[t]\). Theorem 2: Let \(n,B\) be positive integers, \(p\) a prime, \(q\) a power of \(p\), \(f_1(X),\dots,f_r(X)\in{\mathbb F}_q[X]\) non-associate irreducible polynomials such that \(\sum\deg(f_i)\leq B.\) Assume that \(n\) is odd if \(p=2\). Then the number of degree \(n\) monic polynomials \(g(t)\in {\mathbb F}_q[t]\) for which all \(f_i(g(t))\) are irreducible in \({\mathbb F}_q[t]\) is \[ \frac{q^n}{n^r} + O_{n,B}(q^{n-\frac 12}). \]