Homogeneous additive congruences. (English) Zbl 1498.11105
Summary: For prime power \(p^n\), let \(\Theta (k, p^n)\) denote the minimal \(s\) such that for any integers \(a_i\) coprime to \(p\) the congruence \[a_1 x_1^k + a_2 x_2^k + \cdots + a_s x_s^k \equiv 0\;(\mathrm{mod}\;p^n)\] has a primitive solution (\(p\nmid x_i\) for some \(i\)), and \(\Theta^* (k, p^n)\) the minimal \(s\) such that there is always a solution with \(p\nmid x_i\) for all \(i\), \(1 \leq i \leq s\). We obtain a number of estimates for these two quantities when the group \(A\) of \(k\)-th powers mod \(p\) has at least two elements, including, \(\Theta (k, p) \ll \log^3 k, \Theta (k, p) \ll \log^{5 / 2} k\) if \(|A|\) is composite, \(\Theta (k, p^n) \ll k^{\sqrt{5} / \sqrt{\log_2 k}} \log^2 k, \Theta^* (k, p^n) \leq 8k^{5 / \log_{2} (t_{1} /C)}\) for some absolute constant \(C\), where \(t_1 = (p - 1) / (p - 1, k) > C\), and \(\Theta^* (k, p) \leq 8\) for \(p > 6k^2\).
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