Hua-type iteration for multidimensional Weyl sums (Q2902656)

This paper is concerned with counting solutions of a mildly generalized version of the multidimensional Vinogradov mean-value system. Let positive integers \(s,k\) and \(d\) be given. If \(\mathbf{x}\) and \(\mathbf{j}\) are \(d\)-dimensional vectors, where \(\mathbf{j}\) has non-negative integer components, we write \(\mathbf{x}^{\mathbf{j}}=x_1^{j_1}\dots x_d^{j_d}\). Then if \(c_{i,\mathbf{j}}\) are non-zero integer constants for \(1\leq i\leq s\) we write \(N_{s,k,d}(P)\) for the number of integer solutions to the system of equations NEWLINE\[NEWLINEc_{1,\mathbf{j}}\mathbf{x}_1^{\mathbf{j}}+\dots+ c_{s,\mathbf{j}}\mathbf{x}_s^{\mathbf{j}}=0\;\;\;(j_1+\dots+j_d=k)NEWLINE\]NEWLINE with \(\mathbf{x}_i\in[-P,P]^d\). This is a system of NEWLINE\[NEWLINE\ell=\left(\begin{matrix} k+d-1\\ d-1\end{matrix}\right)NEWLINE\]NEWLINE equations in \(sd\) variables.NEWLINENEWLINEThe circle method then leads to a formula NEWLINE\[NEWLINEN_{s,k,d}(P)=cP^{sd-k\ell}+O(P^{sd-k\ell-\delta})NEWLINE\]NEWLINE with \(\delta>0\) depending on \(s,k\) and \(d\), whenever \(s\) is large enough in terms of \(k\) and \(d\). Here \(c\) is the familiar product of local densities. For \(k=3\) and \(d=2\) the problem corresponds to counting lines on a cubic hypersurface in \(\mathbb{P}^{s-1}\), and it is shown that \(s\geq 29\) suffices, improving earlier work of the author [Trans. Am. Math. Soc. 352, No. 11, 5045--5062 (2000; Zbl 1108.11302)]. When \(d=2\) and \(k\geq 4\) it suffices to have \(s\geq k(k-1)2^{k-2}+2k(k+1)+1\), while for \(k=3\) and \(d\geq 2\) it is enough that NEWLINE\[NEWLINEs\geq\min\{2d^3+6d^2-20d+29,\,\tfrac{5}{3}d^3+5d^2+\tfrac{10}{3}d+1\}.NEWLINE\]NEWLINE The proofs involve the development of bounds for exponential sums in the spirit of Weyl, and of mean-value estimates of Hua's type.