Singularities of fractions and behaviour of polynomials at V.I.Arnold \Lambda Summary: Singularities of fractions and behaviour of polynomials at infinity V.I.Arnold \Lambda 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. Collections: Mathematics