On simultaneous additive equations. III. (English) Zbl 0691.10008
Let \(c_ i\) and \(d_ i\) (1\(\leq i\leq s)\) be rational integers, and \(k\geq n>1\) be natural numbers. Consider the pair of additive equations:
\[
(*)\quad c_ 1x^ k_ 1+...+c_ sx^ k_ s=0;\quad d_ 1x^ n_ 1+...+d_ sx_ s^ n=0.
\]
When \(s>2(k+n)\), and p is a rational prime with \(p>k^ 4n^ 2\), it is shown that the equations (*) are simultaneously soluble non-trivially in p-adic integers. This generalizes a result due to O. D. Atkinson and R. J. Cook for the case \(k=n\) [J. Aust. Math. Soc., Ser. A 46, No.3, 438-455 (1989; Zbl 0676.10014)]. An example shows that the condition on s cannot be relaxed. There are similar results when \(n=1.\)
The proof goes via refinements and generalizations of methods previously adopted in part I of this series [Proc. Lond. Math. Soc., III. Ser. (to appear); see the preceding review.]
The proof goes via refinements and generalizations of methods previously adopted in part I of this series [Proc. Lond. Math. Soc., III. Ser. (to appear); see the preceding review.]
Reviewer: T.D.Wooley
MSC:
11D88 | \(p\)-adic and power series fields |
11D72 | Diophantine equations in many variables |
11D41 | Higher degree equations; Fermat’s equation |
Citations:
Zbl 0676.10014References:
[1] | Schmidt, Equations over finite fields. An elementary approach. Lecture Notes in Mathematics (1976) · Zbl 0329.12001 · doi:10.1007/BFb0080437 |
[2] | Greenberg, Lectures on forms in many variables (1969) · Zbl 0185.08304 |
[3] | Dodson, Phil. Trans. Roy. Soc. Lond 261 pp 163– (1966) · Zbl 0139.27102 · doi:10.1098/rsta.1967.0002 |
[4] | DOI: 10.1017/S1446788700030925 · doi:10.1017/S1446788700030925 |
[5] | Davenport, Symp. in Pure Math pp 74– (1967) |
[6] | DOI: 10.1098/rsta.1966.0060 · Zbl 0227.10038 · doi:10.1098/rsta.1966.0060 |
[7] | Chowla, Kon. Norske Vidensk. Selsk. Fork 32 pp 74– (1959) |
[8] | DOI: 10.1098/rsta.1969.0035 · Zbl 0207.35304 · doi:10.1098/rsta.1969.0035 |
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