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Let $0\neq f\in k[X_0,\ldots,X_n]$ be an irreducible polynomial with coefficients in the algebraically closed field $k$. I'm trying to check that there exist $n$ points $x_1,\ldots,x_n\in k$ such that $0\neq f(X_0,x_1,\ldots,x_n)$ has no zeroes with multiplicity greater thatn $1$. Any hint?

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    $\begingroup$ There is just one polynomial in your post. What do you want it to not have common factors with? $\endgroup$ – Lukas Kofler May 13 '18 at 23:17
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    $\begingroup$ By "common factors" do you mean zeroes of multiplicity greater than one? If not that, what? Cheers! $\endgroup$ – Robert Lewis May 13 '18 at 23:23
  • $\begingroup$ Yes @RobertLewis. I mean zeroes of multiplicity$\geq 1$. $\endgroup$ – Vincenzo Zaccaro May 13 '18 at 23:38
  • $\begingroup$ We have complitely right! The question was not well written! $\endgroup$ – Vincenzo Zaccaro May 13 '18 at 23:40

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