The action of a Lie pseudogroup $\mathcal{G}$ on a smooth manifold$M$induces a prolonged pseudogroup action on the jet spaces$J$^{$n$}of submanifolds of$M$. We prove in this paper that both the local and global freeness of the action of $\mathcal{G}$ on$J$^{$n$}persist under prolongation in the jet order$n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

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For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with coordinates in the topological closure of this subgroup in the product of the multiplicative group of those local completions of this function field over all but finitely many places.

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No abstract uploaded!