×

On simultaneous diagonal equations and inequalities. (English) Zbl 0774.11015

Let \(F_ 1,\dots, F_ R\) be forms of degree \(k\) with integer coefficients in \(N\) variables. H. Davenport and D. J. Lewis [Philos. Trans. R. Soc. Lond., Ser. A 261, 97-136 (1966; Zbl 0227.10038)] showed that for odd \(k\) the condition \(N\geq [9R^ 2 k\log(3Rk)]\) assures the existence of a non-trivial solution x of \(F_ 1(\mathbf{x})=\cdots=F_ R(\text\textbf{x})=0\). When \(k\geq 4\) is even they needed \(N\geq [48R^ 2 k^ 3\log(3 Rk^ 2)]\) and an additional “rank condition” on the matrix of coefficients.
In this paper, the authors consider a diagonal system \(G_ i({\mathbf x})=\sum_{j=1}^ N \lambda_{ij} x_ j^ k\), \(1\leq i\leq R\) where \(\lambda_{ij}\in\mathbb{Z}\) and obtain some improvements on the conditions for \(N\), namely, the bounds for \(N\) can be taken growing linearly in \(R\). For instance, suppose that \[ G_ 1(\mathbf{x})=\cdots =G_ R(\text\textbf{x})=0 \tag{1} \] has a non-singular solution in the \(p\)-adic field and that some “rank conditions” on \(\lambda_{ij}\) are satisfied. Then the system (1) has non-trivial solutions if \(N>n_ 0 R\) where \(n_ 0=2k(\log k+O(\log\log k))\) when \(k\) is large. They also obtain a lower bound for the number of solutions of (1).
If \(\lambda_{ij}\in\mathbb{R}\), they obtain parallel results for a system of diagonal inequalities \(| G_ i(\mathbf{x})|<\tau\), \(1\leq i\leq R\).

MSC:

11D72 Diophantine equations in many variables
11P55 Applications of the Hardy-Littlewood method
11D88 \(p\)-adic and power series fields
11D75 Diophantine inequalities

Citations:

Zbl 0227.10038
PDFBibTeX XMLCite
Full Text: DOI EuDML