| Ամփոփում: | Let F(x) =F[x1,...,xn]∈ℤ[x1,...,xn] be a non-singular form of degree d≥2, and let N(F, X)=#{xεℤ n ;F(x)=0, |x|≤X}, where {Mathematical expression}. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces, Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] that N(F, X)≪X n-2+2/n for any fixed form F. It is shown here that the exponent may be reduced to n - 2 + 2/(n + 1), for n ≥ 4, and to n - 3 + 15/(n + 5) for n ≥ 8 and d ≥ 3. It is conjectured that the exponent n - 2 + ε is admissable as soon as n ≥ 3. Thus the conjecture is established for n ≥ 10. The proof uses Deligne's bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the 'q-analogue' of van der Corput's AB process. © 1994 Indian Academy of Science.
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