A Nash function on an open semialgebraic set in $\mathbb{R}^n$ is a semialgebraic function which is smooth. The ring of germs of Nash functions is isomorphic to the Henselisation of $X=\mathbb{R}[x_1,\dots, x_n]$. This is by the way the étale localisation of the latter ring and is just the ring of algebraic power series. Blow-Nash is arc-analycity so one would have a map to the pushforward of X via the map $\pi$ in the étale topology over the real. But this has very bad properties because of some lemma buried in SGA 4 which I forgot. Scheiderer tried to construct a better substitute for this. A sheaf theory for arc-analytic maps would be interesting.

A Nash function on an open semialgebraic set in $\mathbb{R}^n$ is a semialgebraic function which is smooth. The ring of germs of Nash functions is isomorphic to the Henselisation of the localisation of $X=\mathbb{R}[x_1,\dots, x_n]$. This is by the way the étale localisation of the latter ring and is just the ring of algebraic power series. Blow-Nash is arc-analycity so one would have a map to the pushforward of X via the map $\pi$ in the étale topology over the real. But this has very bad properties because of some lemma buried in SGA 4 which I forgot. Scheiderer tried to construct a better substitute for this. A sheaf theory for arc-analytic maps would be interesting.