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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.