Denote by ν(a1, . . . , an) the number of non-degenerate solutions of (1.1). First, let a1, . . . , an be non-zero rational numbers. In 1965, Mann [2] showed that if (ζ1, . . . , ζn) is a non-degenerate solution of (1.1), then ζ 1 = · · · = ζ n = 1, where d is a product of distinct primes ≤ n+1. From this result it can be deduced that ν(a1, . . . , an) ≤ e1 2 for some absolute constant c1. Later, Conway and Jones [1] showed that for every non-degenerate solution (ζ1, . . . , ζn) of (1.1) one has ζ 1 = · · · = ζ n = 1, where d is the product of distinct primes p1, . . . , pl with ∑l i=1(pi−2) ≤ n−1. This implies that ν(a1, . . . , an) ≤ e2 3/2(log n)1/2 for some absolute constant c2. Schinzel [3] showed that if a1, . . . , an are non-zero and generate an algebraic number field of degree D, then ν(a1, . . . , an) ≤ c2(n,D) for some function c2 depending only on n and D. Later, Zannier [5] gave a different proof of this fact and computed c2 explicitly. Finally, Schlickewei [4] succeeded to derive an upper bound for the number of non-degenerate solutions of (1.1) depending only on n for arbitrary complex coefficients a1, . . . , an.