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… 

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References

SHOWING 1-10 OF 10 REFERENCES
Cubic forms in thirty-two variables
  • H. Davenport
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1959
It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at
On the Hasse principle for cubic surfaces
It was conjectured by Mordell [6] that the Hasse principle holds for cubic surfaces in 3-dimensional projective space other than cones†: i.e., that such a surface defined over the rational field 0
A Note on the Diophantine Equation x 3 + y 3 + z 3 = 3
Any integral solution of the title equation has x =y z (9). The report of Scarowsky and Boyarsky [3] that an extensive computer search has failed to turn up any further integral solutions of the
Solutions of the Diophantine equation
. Let ( F n ) n ≥ 0 be the Fibonacci sequence given by F 0 = 0 , F 1 = 1 and F n +2 = F n +1 + F n for n ≥ 0. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a
A note on diophantine equation
In this note we prove that the equation k 2
A note on the Diophantine equation ⁿ+ⁿ+ⁿ=3
Solutions of the Diophantine Equation: x3+y3+z3=k