Result: On representations of positive integers as a sum of two polynomials

Let \(p_ 1(\cdot)\), \(p_ 2(\cdot)\) denote two polynomials of degree \(k\geq 2\) with real coefficients, which are strictly monotone increasing in \([0,\infty[\). A more general version of the well-known Gaussian circle-problem can be represented by the lattice point problem \(R(x):=\#\{(u,v)\in\mathbb{N}^ 2\): \(p_ 1(u)+p_ 2(v)\leq x\}\), where \(x>0\) denotes a large real variable. The authors' interesting asymptotic result is as follows: \[ \begin{aligned} R(x) &= C_ 1 x^{2/k} -C_ 2 x^{1/k}+E(x),\\ E(x) &=F(x)x^{{1/k}-{1/k^ 2}}+O(x^{7/11k} (\log x)^{45/22})+O(x^{(k-1)/k(k+2)}),\tag{*} \end{aligned} \] with positive constants \(C_{1,2}\) depending on \(k\), \(p_{1,2}(\cdot)\) and a uniform convergent sine-series \(F(\cdot)\), which is also \(\Omega(1)\). Note that for \(k\geq 3\) the first term in (*) dominates the two order-terms, thus the exact order of the ``lattice rest'' is determined. The proof itself is based on an application of the so-called ``Discrete Hardy-Littlewood method'' in the form as given by \textit{W. Müller} and \textit{W. G. Nowak} [Lect. Notes Math. 1452, 139-164 (1990; Zbl 0715.11054)].