Lattice points below algebraic curves (Q1915336)

Let \(g(x,y)= \sum_{i+j=d} a_{i,j}x^iy^j\) be a polynomial of degree \(d\geq 3\) with nonnegative integer coefficients \(a_{i,j}\), such that \(a_{d,0}a_{0,d}\neq 0\) and \(g(x,y)\) is not the sum of the \(d\)th power of a linear term and a polynomial of degree less than \(d\). The author studies the average order of the arithmetic function \(r_g(n)\), which counts the number of ways to represent a natural number \(n\) by values of \(g(x,y)\): \[ r_g(n)= \#\{(x,y)\in N^2: g(x,y)=n\}. \] This can be interpreted geometrically as the task of counting the number of lattice points in a large planar domain, the very special case \(g(x,y)= x^2+y^2\) corresponding to the famous Gaussian circle problem. The author's main result reads \[ \sum_{n\leq t} r_g(n)= E_0t^{2/d}+ E_1t^{1/d}+ O(t^{1/d-1/d^2}), \] with real constants \(E_0>0\) and \(E_1\). Under the additional restriction that all coefficients \(a_{i,d-i}\) vanish for \(i=1,\dots,d-1\), the \(O\)-term is replaced by an explicit series representation and a remainder of smaller order. As the author remarks, the estimates are not quite as sharp as those of \textit{G. Kuba} and \textit{W. G. Nowak} [Arch. Math. 58, 147-156 (1992; Zbl 0767.11044)] and \textit{G. Kuba} [Arch. Math. 62, 207-215 (1994; Zbl 0849.11079)] who dealt with the special case that \(g(x,y)= p_1(x)+ p_2(y)\). The method of proof is based on the ``discrete Hardy-Littlewood-method'' for the estimation of exponential sums, in the version of \textit{M. N. Huxley} [Proc. Lond. Math. Soc., III. Ser. 60, 471-502 (1990; Zbl 0659.10057)], along with deep classic tools from complex function theory, like Rouché's theorem and results on algebraic functions.