An analogue of the Mahler-Sprindzhuk theorem for the fourth degree polynomials in two variables (Q1102324)

Continuing his former investigations the author proves the analogue of the Mahler-Sprindzhuk theorem for the polynomials in \((x,y)\) of the fourth degree. Let \[ P(x,y) = \sum _{i,j;0 \leq i + j \leq 4} a _{ij}x^ i y^ j \] be a polynomial with rational integral coefficients, \(h = \max _{(i,j)} |a_{ij}|\). The following two theorems are valid: Theorem 1. Let m be the number of non-zero coefficients among \(a_{ij}\) with \(i + j = 4\) and \(a_{40}\cdot a_{04} = 0\). If \(\epsilon > 0\) is any fixed number, then for almost all real \((x,y)\) the inequality \(|P(x,y)| < h^{-m-9-\epsilon}\) has only a finite number of solutions in the polynomials \(P(x,y)\). Theorem 2. If \(\epsilon > 0\) is any fixed number, then for almost all real \((x,y)\) the inequality \(|P(x,y)| < h^{-15-\epsilon}\) has only a finite number of solutions in the polynomials \(P(x,y)\).