The density of zeros of forms for which weak approximation fails

Author:
D. R. Heath-Brown

Journal:
Math. Comp. **59** (1992), 613-623

MSC:
Primary 11G35; Secondary 11D25, 11P55

DOI:
https://doi.org/10.1090/S0025-5718-1992-1146835-5

MathSciNet review:
1146835

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The weak approximation principle fails for the forms , when 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.

**[1]**J. W. S. Cassels,*A note on the Diophantine equation*, Math. Comp.**44**(1985), 265-266. MR**771049 (86d:11021)****[2]**J. W. S. Cassels and M. J. T. Guy,*On the Hasse principle for cubic surfaces*, Mathematika**13**(1966), 111-120. MR**0211966 (35:2841)****[3]**H. Davenport,*Cubic forms in thirty-two variables*, Philos. Trans. Roy. Soc. London Ser. A**251**(1959), 193-232. MR**0105394 (21:4136)****[4]**V. L. Gardiner, R. B. Lazarus, and P. R. Stein,*Solutions of the Diophantine equation*, Math. Comp.**18**(1964), 408-413. MR**0175843 (31:119)****[5]**R. C. Vaughan,*The Hardy-Littlewood circle method*, Cambridge Univ. Press, 1981. MR**628618 (84b:10002)**

Retrieve articles in *Mathematics of Computation*
with MSC:
11G35,
11D25,
11P55

Retrieve articles in all journals with MSC: 11G35, 11D25, 11P55

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1146835-5

Keywords:
Cubic surfaces,
weak approximation,
Brauer-Manin obstruction,
Hardy-Littlewood formula,
asymptotic estimates

Article copyright:
© Copyright 1992
American Mathematical Society