Le compl\'ementaire des puissances $n$-i\`emes dans un corps de nombres est un ensemble diophantien

Abstract

Given a number field $k$ and a positive integer $n$, there exists an algebraic variety $X$ over $k$ and a function $f$ on $X$ whose set of values $f(X(k))$ on the set of $k$-points of $X$ is the complement in $k$ of the set of $n$-th powers. This result had been proved by B. Poonen (2009) for $n$ a power of $2$. For $n$ arbitrary, under Schinzel's hypothesis, it has been given a conditional proof by T. V\'arilly-Alvarado and B. Viray (2012). Instead of Schinzel's hypothesis, we use "Salberger's trick", as developed in papers of Skorobogatov, Swinnerton-Dyer and one of the authors.