A Hardy -function is a function constructed from and using the field operation, exponentiation, logarithm and algebraic functions. Hardy showed that the germs of -functions at infinity form a totally ordered field. He asked the question whether the functional inverse of can be expanded with respect to an asymptotic scale of -functions. We will prove that this is not the case.

Occasions: ICM 1998, Berlin, August 24, 1998

Documents: slides

Note: the same result was proved independently by Marker, Macintyre and van den Dries. A similar result for the functional inverse of was obtained by Shackell.