Power series and smooth functions equivalent to a polynomial

Abstract:

An algebraic criterion is given for a power series in $n$ variables over a field of characteristic 0 to be equivalent to a polynomial in $n - k$ variables over the ring of power series in $k$ variables. For convergent power series over the reals or complexes a geometric interpretation of the criterion is established. An analogous sufficient condition is obtained for germs of smooth functions. Most of the previously known results follow easily from the criterion.