127,966 research outputs found
Plane curves in boxes and equal sums of two powers
Given an absolutely irreducible ternary form , the purpose of this paper
is to produce better upper bounds for the number of integer solutions to the
equation F=0, that are restricted to lie in very lopsided boxes. As an
application of the main result, a new paucity estimate is obtained for equal
sums of two like powers.Comment: 15 pages; to appear in Math. Zei
Quadratic polynomials represented by norm forms
The Hasse principle and weak approximation is established for equations of
the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic
polynomial in one variable and N is a norm form associated to a quartic
extension of the rationals containing the roots of P. The proof uses analytic
methods.Comment: 55 page
The density of rational points on non-singular hypersurfaces, II
For any integers , let be a non-singular hypersurface of degree that is defined over . The main result in this paper is a proof that the number of -rational points on which have height at most satisfies
for any . The implied constant in this estimate depends at most upon and
Integral points on cubic hypersurfaces
Let g be a cubic polynomial with integer coefficients and n>9 variables, and
assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k.
We show that the equation g=0 has infinitely many integer solutions when the
cubic part of g defines a projective hypersurface with singular locus of
dimension <n-10. The proof is based on the Hardy-Littlewood circle method.Comment: 18 page
Counting rational points on quadric surfaces
We give an upper bound for the number of rational points of height at most
, lying on a surface defined by a quadratic form . The bound shows an
explicit dependence on . It is optimal with respect to , and is also
optimal for typical forms .Comment: 29 page
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