Multiple exponential sums over smooth numbers (Q2706827)

In this substantial paper, estimates for multiple exponential sums are developed and applied to a variety of problems. NEWLINENEWLINENEWLINEOne of the author's goals is to establish the existence of non-trivial rational lines on hypersurfaces defined by the vanishing of a diagonal form: \(\sum_{1 \leq i \leq s}c_ix_i^k\), subject to certain conditions, one of which should be that the number \(s\) of variables in the equation need not exceed a realistically small value. In this respect his theorem improves substantially on an approach developed by R. Brauer and later B. J. Birch, which does not yield a reasonably small bound, and guarantees only the existence of linear spaces on the hypersurfaces which, although of reasonably small degree, cannot be guaranteed to be lines. Relatively recently, some calculations of the number of variables needed in an improved approach of this type in the cubic case have been performed by \textit{T. D. Wooley} [Bull. Lond. Math. Soc. 29, 556-562 (1997; Zbl 0910.11013)]. NEWLINENEWLINENEWLINEThe author uses the Hardy-Littlewood method, which also leads to other applications of a type more commonly seen in this context. Let \(S_s(P)\) denote the number of solutions of the system NEWLINE\[NEWLINE\sum_{1 \leq m \leq s}x_m^iy_m^j - \overline x_m^i\overline y_m^j=0\;\text{ for} 0 \leq i \leq k\;\text{ and} i+j=kNEWLINE\]NEWLINE in numbers not exceeding \(P\). Then, provided \(s \geq s_1\) for a certain \(s_1\) not very different from \(k^2\), the author shows that \(S_s(P) \ll P^{4s-k(k+1)+\Delta_s+\varepsilon}\) with an ``admissible exponent'' not exceeding \(k(k+1)\) and decreasing as \(s\) increases. The author can restrict his variables to be \(R\)-smooth (having no prime factors exceeding \(R\)) with \(R \leq P^\eta\), where \(\eta\) depends on \(\varepsilon\). For large \(k\) the result is sharpened to a form with improved admissible exponents by a process of repeated efficient differencing. These results also lead in a standard way to estimates of Weyl sums over minor arcs. NEWLINENEWLINENEWLINEIn the related problem in which the restriction \(i+j=k\) is absent \textit{G. I. Arkhipov}, \textit{A. A. Karatsuba} and \textit{V. N. Chubarikov} [Tr. Mat. Inst. Steklova 151 (1980; Zbl 0441.10037); translation in Proc. Steklov Inst. Math. 151 (1982)] had an admissible exponent more like \(k^3\) and needed \(s_1\) of a size like \(6k^3\log k\). NEWLINENEWLINENEWLINEThe author also considers a multidimensional analogue of Waring's problem, introduced by \textit{G. I. Arkhipov} and \textit{A. A. Karatsuba} [Sov. Math., Dokl. 36, No. 1, 75-77 (1987); translation from Dokl. Akad. Nauk SSSR 295, 521-523 (1987; Zbl 0655.10043)]. Here he seeks the number of solutions of a system NEWLINE\[NEWLINE\sum_{1 \leq m \leq s}x_m^{k-j}y_m^j=n_j \text{ for } 0 \leq j \leq kNEWLINE\]NEWLINE with the integers \(x_m,y_m\) in \([1,P]\). A condition on the numbers \(n_j\) has to be imposed, sufficient to ensure that real solutions exist. In addition, a \(p\)-adic condition of the usual type is required. Then \(\gg P^{2s-k(k+1)}\) solutions are found provided \(s \geq s_1\), for a certain \(s_1\) somewhat larger than \({14\over 3}k^2\log k\). NEWLINENEWLINENEWLINEIn a related way the author produces a similar result for the system of equations NEWLINE\[NEWLINE\sum_{1 \leq m \leq s}c_mx_m^{k-j}y_m^j=n_j \text{ for } 0 \leq j \leq k,NEWLINE\]NEWLINE or equivalently, via the binomial theorem, of the equation \(\sum_{1 \leq m \leq s} c_m(x_mt+y_m)^k=0\). The connection with the problem of finding lines on a surface is not immediate, because of the possibility that many of the solutions that have been counted might correspond to the same line. This obstacle is overcome by producing an estimate when the variables are restricted to dyadic intervals, and showing that the number of solutions corresponding to a given line is bounded. In this way the author finds, subject to the usual solubility conditions, \(\gg P^{2s-k(k+1)}\) distinct rational lines on the surface \(\sum_{1 \leq m \leq s}c_mz_m^k =0\) with \(\max\{|x_m|,|y_m|\}\leq P\). NEWLINENEWLINENEWLINEThe author announces that in a separate publication [Trans. Am. Math. Soc. 352, 5045-5062 (2000; Zbl 1108.11302)] it is shown that in the cubic case such a result holds unconditionally whenever \(s \geq 57\).