A Note on Symmetric Diagonal Equations

The object of this note is to establish an estimate for the number of solutions of certain simultaneous symmetric diagonal equations which is, in some sense, close to best possible. We shall suppose in what follows that s and t are positive integers, that k_j (1 ≤ j ≤ t) are positive integers with 1 ≤ k ₁ < k ₂ < ⋯ < k_t , and that P is a large positive real number. Define S_s (P; k) to be the number of solutions of the simultaneous diophantine equations (1) ∑ i = 1 s ( x i k j   −   y i k j )   =   0     ( 1   ≤   j   ≤   t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2642.tif"/>