On \(p\)-adic zeros of forms (Q788021)

The author sharpens the remarkable result of \textit{G. I. Arkhipov} and \textit{A. A. Karatsuba} [Math. USSR, Izv. 19, 231--240 (1982); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45, 948--961 (1981; Zbl 0476.10018)] that the number of variables possible in a form having only the trivial p-adic zero has an exponential lower bound with respect to the degree (and so is not even limited to polynomial growth). His estimate is stronger than Arkhipov and Karatsuba's recently announced improvement [Usp. Mat. Nauk 37, No. 5(227), 161--162 (1982; Zbl 0508.10012)] of their original results; their improvement is in turn stronger than the result of \textit{D. J. Lewis} and \textit{H. L. Montgomery} [Mich. Math. J. 30, 83--87 (1983; Zbl 0531.10026)].