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The classical Lê-Greuel formula, proved independently by G.-M. Greuel and Lê Dũng Tráng says that if f1, · · · , fk and g are holomorphic map germs (C, 0) → (C, 0) such that f = (f1, · · · , fk) and (f, g) define isolated complete intersection germs (ICIS for short), then the Milnor number μ(f, g) of (f, g) can be computed in terms of that of f by the formula: μ(f, g) + μ(f) = dimC On+k,0 (f,Jac(f,g)) , where Jac(f, g) denotes the ideal generated by the determinants of all the (k + 1) minors of the corresponding Jacobian matrix. The term on the right can also be regarded as the number of critical points of a Morsification of g on a Milnor fibre of f . In this work we extend this theorem to the rather general setting of a function f = (f1, · · · , fk) : X → C defined on a complex analytic variety X with arbitrary singular set, and another function g : X → C. We impose some conditions on these functions in order to have Milnor, or rather Milnor-Lê, type fibrations. We ask f to be generically a submersion with respect to some Whitney stratification on X, the dimension of its zero set V (f) must be larger than 0, and f must have the Thom property with respect to this stratification. Then we ask g to have an isolated critical point in the stratified sense, both on X and on V (f) of f . We show that the formula of Lê and Greuel extends to the general setting described above. We also restrict to the case when the singularities of X are all contained in V , so that the fibers Ff and Ff,g are non-singular. We get, in particular, an extension of that formula to the case of complete intersection germs in C which does not require that f defines an ICIS, f can be an arbitrary complete intersection germ, provided it has the Thom property and g has an isolated singularity.

The Lê-Greuel formula for functions on analytic spaces