The cubic case of the main conjecture in Vinogradov's mean value theorem (Q266136)

For positive integers \(s\) and \(k\), we denote by \(J_{s,k}(X)\) the number of integral solutions of the system NEWLINE\[NEWLINE \left\{x_1^j+\dots+x_s^j=y_1^j+\dots+y_s^j\right\}_{1\leq j\leq k}, NEWLINE\]NEWLINE with \(1\leq x_i,y_i\leq X\). The main conjecture in Vinogradov's mean value theorem asserts that \(J_{s,k}(X)\ll X^\varepsilon(X^s+X^{2s-\frac{1}{2}k(k+1)})\) for each \(\varepsilon>0\). The conjecture is trivial for the cases \(k=1,2\). In the paper under review, the author gives the first proof of this conjecture in a case with \(k>2\), by proving that NEWLINE\[NEWLINE J_{s,3}(X)\ll X^\varepsilon\left(X^s+X^{2s-6}\right), NEWLINE\]NEWLINE for each \(\varepsilon>0\). Among some applications of the method of proving this result, the author obtains true order of \(J_{s,3}(X)\) when \(s>6\), and considers a diagonal Diophantine system consisting of a cubic, quadratic and linear equation.