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Le complémentaire des puissances
$n$-ièmes dans un corps de nombres est un ensemble diophantien
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 10 / October 2015
- Published online by Cambridge University Press:
- 22 June 2015, pp. 1965-1980
- Print publication:
- October 2015
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- Article
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For
$n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For 
$n=p$, 
$p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its 
$(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).
