On the number of solutions of an $n$th degree congruence with one unknown

We show that the number of solutions to the congruence $f(x)\equiv 0\pmod m$, where $f(x)$ is a polynomial of degree $n\geqslant2$ whose coefficients have greatest common divisor relatively prime to $m$, does not exceed $(n/e+O(\ln^2 n))m^{1-1/n}$, where $n/e+O(\ln^2n)$ cannot be replaced by $n/e$.
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