The density of zeros of forms for which weak approximation fails

@article{HeathBrown1992TheDO,
title={The density of zeros of forms for which weak approximation fails},
author={D. R. Heath-Brown},
journal={Mathematics of Computation},
year={1992},
volume={59},
pages={613-623}
}
The weak approximation principal fails for the forms x + y + z = kw, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these forms. Evidence, both numerical and theoretical, is presented, which suggests that, for forms of the above type, the product of the local densities still gives the correct global density. Let f(x1, . . . , xn) ∈ Q[x1, . . . , xn] be a rational form. We say that f satisfies the weak approximation…
32 Citations
1 Hilbert ’ s Tenth Problem
Hilbert’s 10th problem, stated in modern terms, is Find an algorithm that will, given p ∈ Z[x1, . . . , xn], determine if there exists a1, . . . , an ∈ Z such that p(a1, . . . , an) = 0. Davis,
Rational points on smooth cubic hypersurfaces
Let S be a smooth n-dimensional cubic variety over a field K and suppose that K is finitely generated over its prime subfield. It is a well-known fact that whenever we have a set of K-points on S, we
Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction
• N. Elkies
• Mathematics, Computer Science
ANTS
• 2000
A new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C is given, and its proof also yields new estimates on the distribution mod 1 of (cx)3/2 for any positive rational c.
Hilbert's Tenth Problem: Refinements and Variants
Hilbert’s 10th problem, stated in modern terms, is Find an algorithm that will, given p ∈ Z[x1, . . . , xn], determine if there exists a1, . . . , an ∈ Z such that p(a1, . . . , an) = 0. Davis,
Rational points near manifolds and metric Diophantine approximation
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have
N T ] 1 4 M ay 2 00 0 Rational points near curves and small nonzero | x 3 − y 2 | via lattice reduction
We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat
Tamagawa Numbers of Diagonal Cubic Surfaces of Higher Rank
• Mathematics
• 2001
We consider diagonal cubic surfaces defined by an equation of the form \$\$a{{x}^{3}} + b{{y}^{3}} + c{{z}^{3}} + d{{t}^{3}} = 0.\$\$ Numerically, one can find all rational points of height
Computational number theory at CWI in 1970--1994
• Computer Science
• 1994
A concise survey of the research in Computational Number Theory, carried out at CWI in the period 1970 to 1994, with updates to the present state-of-the-art of the various subjects, if necessary.
Integral points on log K3 surfaces
Let X be Z-scheme, i.e., a smooth separated scheme of finite type over Z. In this talk most schemes of interest will be affine, and so given by a collection of polynomial equations with integral
Sum of Three Cubes via Optimisation.
• Mathematics
• 2020
By first solving the equation \$x^3+y^3+z^3=k\$ with fixed \$k\$ for \$z\$ and then considering the distance to the nearest integer function of the result, we turn the sum of three cubes problem into an