Small prime solutions to cubic Diophantine equations (Q2862959)

The author considers in this paper the equation NEWLINE\[NEWLINE n = a_1 p_1^3 + \cdots a_9p_9^3,NEWLINE\]NEWLINE where the \(p_j\)'s are prime variables and the coefficients \(a_j\)'s are pairwise coprime nonzero integers. The local condition for solubility of this equation is \(n \equiv a_1 + a_2 +\cdots + a_9 \pmod 2\). Let \(A = \max \{2, |a_1|, |a_2|, \ldots , |a_9|\}\). The main results proved in this paper are the following:{\parindent=6mm \begin{itemize} \item[(i)] If the coefficients \(a_j\)'s are not all of the same sign, then the above equation has solutions which satisfies \(p_j \ll |n|^{1/3} + A ^{14 +\varepsilon}\).\item [(ii)] If the coefficients \(a_j\)'s are all positive, then the above equation is soluble whenever \( n \gg A^{43 + \varepsilon}\). Here \(\varepsilon >0\) is arbitrary. NEWLINENEWLINE\end{itemize}} This area of problems is an extension of the similar investigation on the small prime solutions of ternary linear equations in three prime variables in which a special case was initiated by \textit{A. Baker} [J. Reine Angew. Math. 228, 166--181 (1967; Zbl 0155.09202)] and then in general form by \textit{M.-C. Liu} and \textit{K.-M. Tsang} [Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 595--624 (1989; Zbl 0682.10043); Monatsh. Math. 111, No. 2, 147--169 (1991; Zbl 0719.11064); Acta Arith. 118, No. 1, 79--100 (2005; Zbl 1081.11065)].NEWLINENEWLINEThe technical novelty in the present paper is an enlargement of the major arcs by introducing a sequence of larger arcs, where on each of these the application of well-known bounds on exponential sums over cubes of primes can be made more efficient. Such results can be compared to the cases of linear ternary equations and additive quadratic equations in five primes variables.