Geodesic
On~local representation of zero by a~form
Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 231-240.

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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 $Q_p$. Bibliography: 6 titles. 
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G. I. Arkhipov; A. A. Karatsuba. On~local representation of zero by a~form. Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 231-240. http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a1/

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