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Nov 29, 2016 at 12:20 vote accept Daniel Loughran
Nov 28, 2016 at 13:58 comment added Martin Bright I guess you can make this a bit neater by using that X is a local complete intersection, so you can take $r=m-n$. Then the Jacobian is surjective, and you don't need to appeal to Hensel's Lemma to show existence of a solution.
Nov 28, 2016 at 12:49 history answered Martin Bright CC BY-SA 3.0