The asymptotics of a lattice point problem associated to a finite number of polynomials. II
[Part I, cf. ibid. 63, 139-192 (1991; Zbl 0735.11048).] The aim of this paper is to study the asymptotic behavior of the number of lattice points in \(\mathbb{R}^2\). Let \(P_1 (x_1, x_2)\), \(P_2 (x_1, x_2)\) be polynomials in \((x_1, x_2)\in \mathbb{R}^2\). Suppose that both are proper mappings on \([1, \infty)^2\) and assume that each is positive at least outside a compact subset of \([1,\infty)^2\). We set, for \(t_1, t_2>0\), \(N(t_1, t_2):= \#\{m\in \mathbb{R}^2\mid P_i (m)\leq t_i\}\). Under suitable conditions, the author proves the following result: For a pair \(v= (v_1, v_2)\), there exists a polynomial \(A_v (u_1, u_2)\) that is positive outside a compact subset of \((1, \infty)^2\) so that, given any subregion \(R_\infty (v)\), a positive number \(\theta\) exists such that \[ N(t_1, t_2)= t_1^{v_1} t_2^{v_2} A_v (\log t_1, \log t_2)+ O(t_1^{v_1- \theta}, t_2^{v_2- \theta}) \] if \((t_1, t_2)\to (\infty, \infty)\) and \((t_1, t_2)\in \mathbb{R}_\infty (v)\). Further a related result is given.
scientific article; zbMATH DE number 784719
scientific article