On representations of positive integers as a sum of two polynomials. II (Q1319585)

[For part I, cf. ibid. 58, 147-156 (1992; Zbl 0767.11044).] This paper studies the average order of the arithmetical function \(r(p, q; \nu)\), which counts the number of representations of a natural number \(\nu\) in the form \(p(u)+ q(v)\) with positive integers \(u\), \(v\) and polynomials \(p,q \subset \mathbb{Z} [x]\) of degree \(2\leq n\leq m\), which are positive and strictly monotone increasing in \([1, \infty[\). In part I the author and \textit{W. G. Nowak} derived an asymptotic formula for the summatory function under the condition that both polynomials have the same degree, which is now relaxed. Especially, for \(m\geq n\geq 3\) one has \[ \sum_{\nu\leq x} r(p, q; \nu)= C_1 x^{(1/n)+(1/m)}- C_2 x^{1/n}- C_3 x^{1/m}+ \begin{cases} O_{p,q} (x^{(1/n)- (1/mn)}),\\ \Omega_{p,q} (x^{(1/n)- (1/mn)}), \end{cases} \qquad (x\to \infty) \] with certain positive constants \(C_1\), \(C_2\), \(C_3\) depending on \(p\), \(q\). The proof itself utilizes \textit{M. N. Huxley}'s discrete Hardy-Littlewood method [Proc. Lond. Math. Soc., III. Ser. 66, 279-301 (1993; Zbl 0820.11060)].