×

Simultaneous equations of additive type. (English) Zbl 0207.35304

In this paper the authors investigate the solubility of a system of \(R\) simultaneous equations of the type
\[ \begin{aligned} a_{11}x_1^k + \ldots + a_{1N}x_N^k &= 0 \\ \vdots \\ a_{R1}x_1^k + \ldots + a_{RN}x_N^k &= 0 \end{aligned} \tag{1} \]
in integers \(x_1, \ldots, x_N\) not all zero. The coefficients \(a_{ij}\) are arbitrary integers. When \(k\) is odd they establish the following result:
Theorem 1. Let \(k\) be an odd positive integer. The equations (1) have a solution in integers \(x_1, \ldots, x_N\) not all zero if \(N\ge [qR^2k\log 3Rk]\). Here \([x]\) denotes the greatest integer contained in \(x\).
Theorem 2 deals with the case when \(k\) is even.
The method used by the authors is the famous so-called Hardy-Littlewood circle method. But an essential preliminary is the determination of conditions which will ensure the solubility of the equations (1) in every \(p\)-adic field. In their discussion of the latter, the authors quote a famous result of Chevalley (see the author’s Lemma 1). In their proof of Lemma 2 the authors say “we use a straightforward extension of an argument of S. Chowla and G. Shimura” [Norske Vid. Selsk. Forhdl. 36 (1963), 169–176 (1964; Zbl 0119.04405)]. These authors discussed the case \(R=1\) of the problem discussed in this paper.

MSC:

11D72 Diophantine equations in many variables
11P55 Applications of the Hardy-Littlewood method

Citations:

Zbl 0119.04405
PDFBibTeX XMLCite
Full Text: DOI