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# SOLUTIONS TO DIAGONAL CONGRUENCES WITH VARIABLES RESTRICTED TO A BOX

Published online by Cambridge University Press:  03 April 2018

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## Abstract

We prove that for any positive integers $k,n$ with $n>\frac{3}{2}(k^{2}+k+2)$ , prime $p$ , and integers $c,a_{i}$ , with $p\nmid a_{i}$ , $1\leqslant i\leqslant n$ , there exists a solution $\text{}\underline{x}$ to the congruence $$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}a_{i}x_{i}^{k}\equiv c\hspace{0.6em}({\rm mod}\hspace{0.2em}p)\end{eqnarray}$$
with $1\leqslant {x_{i}\ll }_{k}p^{1/k}$ , $1\leqslant i\leqslant n$ . This upper bound is best possible. Refinements are given for smaller $n$ , and for variables restricted to intervals in more general position. In particular, for any $\unicode[STIX]{x1D700}>0$ we give an explicit constant $c_{\unicode[STIX]{x1D700}}$ such that if $n>c_{\unicode[STIX]{x1D700}}k$ , then there is a solution with $1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$ .

## MSC classification

Type
Research Article
Information
Mathematika , 2018 , pp. 430 - 444
Copyright
Copyright © University College London 2018

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SOLUTIONS TO DIAGONAL CONGRUENCES WITH VARIABLES RESTRICTED TO A BOX
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