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Le complémentaire des puissances $n$ -ièmes dans un corps de nombres est un ensemble diophantien

  • Jean-Louis Colliot-Thélène (a1) and Jan Van Geel (a2)

Abstract

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).

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References

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Le complémentaire des puissances $n$ -ièmes dans un corps de nombres est un ensemble diophantien

  • Jean-Louis Colliot-Thélène (a1) and Jan Van Geel (a2)

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