An extension of a result of Zaharescu on irreducible polynomials (Q981852)

Let \(K\) be a field equipped with two non-archimedean valuations \(v_1\) and \(v_2\) of arbitrary rank with value groups \(\Gamma_1\) and \(\Gamma_2\). Let \(A\) be a subring of \(K\) with field of fractions \(K\) which is integrally closed in \(K\), \(\tilde{A}\) the integral closure of \(A\) in the algebraic closure \(\tilde{K}\) of \(K\), \(\tilde{v}_1\) and \(\tilde{v}_2\) valuations on \(\tilde{K}\) whose restrictions to \(K\) coincide to \(v_1\) and \(v_2\) respectively. Assume that for any \(\beta\in\tilde{A}\setminus A\) and \(\lambda_2\in\Gamma_2\), there exists an element \(\lambda_1\in\Gamma_1\) such that \[ \tilde{v}_1(u-\beta)\leq \lambda_1\,\text{for all}\,u\in A\,\text{with}\,v_2(u)\geq \lambda_2. \] The authors prove that for any polynomial \(f(x)=x^d+a_1x^{d-1}+\ldots+a_d\in A[x]\) irreducible over \(K\) and any \(\lambda_2\in\Gamma_2\), there corresponds \(\lambda_1\in\Gamma_1\) depending upon \(f\) and \(\lambda_2\) such that for any \(b_1,b_2,\ldots,b_d\in A\) satisfying \[ v_1(b_i-a_i)\geq \lambda_1,\quad 1\leq i\leq d, \] and \[ v_2(b_i)\geq \lambda_2,\quad 1\leq i\leq d, \] the polynomial \(g(x)=x^d+b_1x^{d-1}+\ldots+b_d\) is irreducible over \(K\). This extends a result of \textit{A. Zaharescu} [Hiroshima Math. J. 34, No. 2, 161--176 (2004; Zbl 1062.11075)].