TY - JOUR

T1 - Analytic order of singular and critical points

AU - Shustin, Eugenii

PY - 2004/3

Y1 - 2004/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=1342288880&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-03-03409-3

DO - 10.1090/s0002-9947-03-03409-3

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AN - SCOPUS:1342288880

SN - 0002-9947

VL - 356

SP - 953

EP - 985

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -