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.]