Analytic order of singular and critical points

Eugenii Shustin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 √μ + 1 ≤ d ≤ μ + 1 where μ is the Milnor number. Our result in both the problems is d ≤ a≤√μ 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.

Original languageEnglish
Pages (from-to)953-985
Number of pages33
JournalTransactions of the American Mathematical Society
Issue number3
StatePublished - Mar 2004


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