  We deal with the following closely related problems: (i) For a germ of a
reduced plane analytic curve, what is the minimal degree of an algebraic curve
with a singular point analytically equivalent (isomorphic) to the given one?
(ii) For a germ of a holomorphic function in two variables with an isolated
critical point, what is the minimal degree of a polynomial, equivalent to the
given function up to a local holomorphic coordinate change? Classically known
estimates for such a degree $d$ in these questions are $\sqrt{\mu}+1\le d\le
\mu+1$, where $\mu$ is the Milnor number. Our result in both the problems is
$d\le a\sqrt{\mu}$ with an absolute constant $a$. As a corollary, we obtain
asymptotically proper sufficient conditions for the existence of algebraic
curves with prescribed singularities on smooth algebraic surfaces.
