A metric theorem on the simultaneous approximation of a zero by the values of integral polynomials

In this paper it is proved that the inequality
$$ \prod_{i=1}^k|P(\omega_i)|<H^{-n+k-1-\varepsilon} $$
has only a finite number of solutions in integral polynomials $P(x)$ for almost all $\overline\omega=(\omega_1,\dots,\omega_k)$. Here $H$ is the coefficient of $P(x)$ largest in absolute value. Sprindzuk's conjecture is thereby proved.
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