FINITE DETERMINACY OF FUNCTIONS WITH NON-ISOLATED SINGULARITIES RUUD PELLIKAAN [Received 28 April 1987] Introduction An analytic function / is finitely determined if and only if it has an isolated singularity; this follows from the work of Mather [21] and Tougeron [35], and was generalized by Siersma [33], who considered functions with a line as singular locus. We generalize this theory to functions which have a given analytic space 2 as singular locus. If / is the ideal defining the analytic space 2 in C m , then we look at all functions which have 2 in their singular locus, and call this collection j I. The group % of all local analytic isomorphisms of (C m , 0) which leave (2, 0) invariant acts on J /. The tangent space of the orbit of / under the action of % we call T 7 (/), and we use C/(/) to denote the /-codimension of / i n J /; that is, c 7 (/) = dim c (J//*,(/)). In § 1 we investigate the ideal J / and give conditions which imply that J / = I 2 . In §§ 2 and 3 we look at the group % and compute its tangent space as well as T 7 (/). In § 4 we characterize those functions / which have Cj(f) = 0, in the case where 2 is non-singular. In § 5 we characterize those functions which have finite /-codimension for some ideal /, and show that T 7 (/) = (mJ f ) n J /, where J f is the Jacobi ideal of /. In § 6 we prove the finite /-determinancy theorem and the existence of a versal /-unfolding when c z (/) <<», using the work of Damon [12]. In § 7 we characterize in geometric terms those functions which have finite /-codimension, in the cases where / defines a reduced curve. We show that certain functions with non-isolated singularities are right-equivalent with a polynomial. In § 8 we give an example of a function which is not right-equivalent with a polynomial, but which is contact-equivalent with a polynomial. This answers negatively a question posed by Bochnak and Kucharz [6]. Most of the time 6 denotes the local ring of germs of analytic functions / : (C m , 0)—»C, and m denotes its maximal ideal. But sometimes in proofs we take representatives of germs. Then 6 denotes the sheaf of analytic functions on an open neighbourhood of 0 in C m . For a map F: C * x C - > C " with q- parameters, we denote by/,: C m - » C , the map defined by/,(z) = F(z, t). For an ideal / in 0 we denote by V(I) the germ of the zero set of / in (C w , 0). We delete the summation variable in a summation if this causes no confusion. For basic facts in commutative algebra we refer to Matsumura [22]. Acknowledgement. This work was done at the University of Utrecht and was part of the author's thesis [26]. I want to thank Professor Dr D. Siersma for his guidance and ideas. This work was supported by the Netherlands Foundation for A.M.S. (1980) subject classification: 32B30, 57R45. Proc. London Math. Soc. (3) 57 (1988) 357-382.