Criteria for equivalence between power series and polynomials

Let $A$ be the ring of formal power series in $n$ variables over a field $K$ of characteristic zero. Two power series $f$ and $g$ in $A$ are said to be equivalent if there exists a $K$-automorphism of $A$ transforming $f$ into $g$. In my talk I will review criteria for a power series to be equivalent to a power series which is a polynomial in at least some of the variables. For example, each power series in $A$ is equivalent to a polynomial in two variables whose coefficients are power series in $n - 2$ variables. In particular, each power series in two variables over $K$ is equivalent to a polynomial with coefficients in $K$. Similar results are valid for convergent power series, assuming that the field $K$ is endowed with an absolute value and is complete. In the special case of convergent power series over the field of real numbers some weaker notions of equivalence will be also considered. I will report on works of several mathematicians giving simple proofs. Some open problems will be included. singularities - power series