Counting rational points on hypersurfaces and higher order expansions
نویسندگان
چکیده
منابع مشابه
Counting Rational Points on Hypersurfaces
For any n ≥ 2, let F ∈ Z[x1, . . . , xn] be a form of degree d ≥ 2, which produces a geometrically irreducible hypersurface in P. This paper is concerned with the number N(F ; B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F ; B) = O(B), whenever either n ≤ 5 or the hypersurface is not a union of lines. Here the implied constant depends at ...
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R0<b162R0 gcd(b1, N )1/2 R 0 (HP) . The second line is false and in fact one has M1 = 1 in Proposition 3. The author is very grateful to Professor Hongze Li for drawing his attention to this flaw. The error can be fixed by introducing an average over b1 into the statement of Proposition 3. This allows us to recover the main theorem in [1], and also [2, Lemma 11], via the following modification....
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Let X ⊂ P be a geometrically integral cubic hypersurface defined over Q, with singular locus of dimension 6 dimX − 4. Then the main result in this paper is a proof of the fact that X(Q) contains Oε,X(B ) points of height at most B.
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Let f(x) = x+ax+bx+cx ∈ Z[x] and consider the hypersurface of degree five given by the equation Vf : f(p) + f(q) = f(r) + f(s). Under the assumption b 6= 0 we show that there exists Q-unirational elliptic surface contained in Vf . If b = 0, a < 0 and −a 6≡ 2, 18, 34 (mod 48) then there exists Q-rational surface contained in Vf . Moreover, we prove that for each f of degree five there exists Q(i...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2017
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2016.09.001