Summary: Singularities of fractions and behaviour of polynomials at
1 Classification of singularities of fractions
In this paper we mean by fraction the ratio (ordered pair) of two germs of (holomorphic or
smooth) functions in n variables at a point (which usually will be refered as the origin). For
the sake of simplicity mostly the complex case is considered below. The results for the real
case are similar, only certain formulas contain nonequivalent real normal forms (which differ
usually by \Sigma signes at certain monomials).
The singularities of fractions will be classified up to the following equivalence relations.
Definition 1. Two fractions are called coinciding, if their numerators and denominators are
proportional (by nonzero function).
Example 1. On C
2 with coordinates x and y the fraction x 2 =xy does not coincide neither
with x=y, nor with x 3 =x 2 y, but it coincides with x 2 e x =xye x .
Definition 2. Two fractions are called Requivalent, if one of them becomes coinciding with
the other when transfered by a germ of biholomorphism of the space of variables.
Example 2. The fraction 1=xy is Requivalent to the fraction 1=(x 2 + y 2 ).
Definition 3. Two fractions are called R + equivalent if one of them is Requivalent to the
sum of the other one with a germ of a holomorphic function.