ON THE RELATION BETWEEN INDEX AND MULTIPLICITY ANNA CIMA, ARMENGOL GASULL JOAN TORREGROSA A This paper is mainly devoted to the study of the index of a map at a zero, and the index of a polynomial map over n . For semi-quasi-homogeneous maps we prove that the index at a zero coincides with the index at this zero of its quasi-homogeneous part. For a class of polynomial maps with finite zero set we provide a method which makes easier the computation of its index over n . Finally we relate the index and the multiplicity. 1. Notation and statement of the results Let f :( n , 0) ( n , 0) be a continuous map such that 0 is isolated in f -(0). Then the index ind [ f ] of f at zero is defined as follows: choose a ball B ε about 0 in n so small that f -(0)B ε 0 and let S ε be its boundary (n1)-sphere. Choose an orientation of each copy of n . Then the index of f at zero is the degree of the mapping ( f f ): S ε S, the unit sphere, where the spheres are oriented as (n1)- spheres in n . If f is differentiable, this degree can be computed as the sum of the signs of the Jacobian of f at all the f-preimages near 0 of a regular value of f near 0. If f is a smooth (that is a ) map, then consider the germ f of f at 0, and the local ring ( n )( f ) of f at 0, where ( n ) is the ring of germs at 0 of smooth real-valued functions on n , and ( f ) is the ideal generated by the components of f . The multiplicity μ [ f ] of f at 0 is defined by μ [ f ] dim [ ( n )( f )] and we say that f is a finite map germ if μ [ f ] . It is known that μ [ f ] is the number of complex f-preimages near 0 of a regular value of f near 0. Given a map g :( n , 0) ( n , 0), where g (g , … , g n ) with each g i a homo- geneous polynomial such that 0 is isolated in g -(0), it is well known that μ [g] n i = d i , where d i is the degree of each g i . On the other hand any smooth function f i :( n , 0) (, 0) can be written as f i g i G i , where g i is the first non-zero jet of f i . Hence, any smooth map f :( n , 0) ( n , 0) can be written as f gG. It is also known that μ [ f ] μ [g] if 0 is isolated in g -(0). Sometimes the above construction provides a g such that 0 is not isolated in g -(0), but a suitable selection of weights associated with any variable (see the definitions in the sequel) makes possible a different decomposition f gG satisfying μ [ f ] μ [g]. We begin this paper by giving a similar property but one concerning indices instead of multiplicities. In order to enunciate the result, we need some preliminary definitions. Received 23 May 1995 ; revised 6 February 1996. 1991 Mathematics Subject Classification 57R45. The authors were partially supported by the DGICYT grant number PB96-1153. J. London Math. Soc. (2) 57 (1998) 757–768