4,778 research outputs found
On the representation of integers by quadratic forms
Let Q be a non-singular quadratic form with integer coefficients. When Q is
indefinite we provide new upper bounds for the least non-trivial integral
solution to the equation Q=0. When Q is positive definite we provide improved
upper bounds for the least positive integer k such that the equation Q=k is
insoluble in integers, despite being soluble modulo every prime power.Comment: 33 page
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
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
Sums of arithmetic functions over values of binary forms
Given a suitable arithmetic function h, we investigate the average order of h
as it ranges over the values taken by an integral binary form F. A general
upper bound is obtained for this quantity, in which the dependence upon the
coefficients of F is made completely explicit.Comment: 12 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
Binary forms as sums of two squares and Ch\^atelet surfaces
The representation of integral binary forms as sums of two squares is
discussed and applied to establish the Manin conjecture for certain Ch\^atelet
surfaces defined over the rationals.Comment: 33 page
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