On local representation of zero by a form

In this article it is proved that for any natural number $n\geqslant n_0$ and for any $p$ there exists a form $F$ of degree not exceeding $n$ whose coefficients are integral over $Q_p$ and whose number $k$ of variables satisfies the inequality
$$ k\geqslant p^u,\qquad u=\frac n{\log_p^2n\log_p\log_p^3n},\quad\log_p\log_p\log_p\log_p\log_p\log_p n_0=11, $$
which can only trivially represent zero in $Q_p$.
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