On~local representation of zero by a~form

G. I. Arkhipov ; A. A. Karatsuba

Izvestiya. Mathematics , Tome 19 (1982) no. 2, pp. 231-240

Résumé

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.