Milnor Algebras and Equivalence Relations among Holomorphic Functions

Let On+i denote the ring of germs at the origin of holomorphic functions (C ,0) —• C. As a ring 0n+i has a unique maximal ideal ra, the set of germs of holomorphic functions which vanish at the origin. Let Gn+\ be the set of germs at the origin of biholomorphisms <\>: (C ,0) —• (C ,0). The following are three fundamental equivalent relations in 0n+iDEFINITION 1. Let ƒ, g be two germs of holomorphic functions (C ,0) —• (C,0). (i) ƒ is right equivalent to g if there exists a 0 G Gn+i such that ƒ' = go(f>. (ii) ƒ is right-left equivalent to g if there exists a 0 G Gn+i and ip G G\ such that ƒ = ip o g o (j). (iii) ƒ is contact equivalent to g if (V, 0) is biholomorphic equivalent to (VV, 0) where V = {z G C n + 1 : ƒ(*) 0} and W = {z G C n + 1 : g(s) = 0}, i.e., there exists a G Gn+i such that 0: (V,0) —• (W,0). One of the natural and fundamental problems in complex analytic geometry is to tell when two germs of holomorphic functions (C ,0) —• (C,0) are equivalent in the sense of (i), (ii), or (iii) respectively in Definition 1. To answer the above problem, we need the following notations: