Let \(a_1,\dots,a_s\) be any non-zero integers and \(k\) be any positive integer. W. Schmidt obtained [Acta Math. 143, 219-232 (1979; Zbl 0458.10020)] that for any \(\varepsilon>0\) there exists a positive constant \(C(k,\varepsilon)\) depending on \(k\) and \(\varepsilon\) only such that if \(s\geq C(k,\varepsilon)\), then the diagonal equation \[ \sigma_1a_1x_1^k+\cdots+\sigma_sa_sx_s^k=0 \] has a nontrivial integer solution in \(\sigma_1,\dots,\sigma_s\); \(x_1,\dots,x_s\) satisfying \(\sigma_j=\pm 1\) and \(| x_j|\leq A^\varepsilon\), \(j=1,\dots,s\) where \(A=\max_{1\leq j\leq s}| a_j|\).
In the present paper the author gives some quantitative results on upper bounds for \(C(k,\varepsilon)\) as follows:
(i) If \(\log A\leq 1/\varepsilon\) then \(C(k,\varepsilon)\leq\max\{2/\varepsilon,20\}\).
(ii) If \(\log A>1/\varepsilon\) then \[ C(k,\varepsilon)\leq c_1c_2^p\begin{cases} 1\quad & \text{if }| a_j|\geq A/2\text{ for }j=1,\dots,s,\\ [4/\varepsilon]\quad & \text{otherwise,}\end{cases} \] where \[ c_1=\begin{cases} 2^k+1\quad & \text{for }2\leq k\leq 11\\ [5k^2\log k]\quad & \text{for }k\geq 12\end{cases},\quad c_2=100c_1k^22^k+c_1^2 \] and \(p=2[\log(1c_1/\varepsilon)]\).