T. Fukui introduced an invariant for the blow-analytic equivalence of real singularities. For a nondegenerate analytic function (germ) f, he discovered a formula for computing the one-dimensional invariant, denoted by A(f) := A1(f). We find a formula for A(f) for any f (real or complex, degenerate or not). We then define, and characterise, various notions of stability of A(f), using the formula. For real analytic f, the Fukui invariant with sign is defined, and computed by a similar formula. In the case where f is an analytic function of two complex variables, A(f) can also be computed using the tree-model of f.

The well-known Tschirnhausen transformation, , eliminates the second term of the polynomial xn + axn-l + …. By a mere repeated application of this transformation, one can decide whether a given element of k[[x,y]] is prime (irreducible) or not. Here K is an algebraically closed field of characteristic 0. A generalised version of Hensel's Lemma is developed for the proofs. The entire paper can be understood by undergraduate students.

When a pair of adjacent C1 strata (X, Y) is Whitney (a)- regular at a point , there is, up to homeomorphism, just one germ at 0 of intersections S ∩ X where S is a Lipschitz transversal to Y at 0 (S is the graph of a Lipschitz map from The proof uses a blowing-up construction and a strong form of Verdier's (w)-regularity.

The Newton polygon and the Newton-Puiseux algorithm ([3], p. 370, [8], p. 98), and their generalizations, serve as a powerful tool for analysing the singularities of a given function. Yet experts know how difficult it is to keep track of them when one, or several, blowing-ups are applied. Thus many interesting theorems are stated under the strong, rather undesirable, assumption that the Newton faces are non-degenerate.

In this paper, we introduce a method which is parallel to the classical Newton-Puiseux theory, yet avoids blowing-ups and fractional power series, except in the proofs.

Let (respectively ) denote the ring of germs at 0 ∈ Rn of all C∞ functions (respectively Cμ functions) from Rn to R. For a given where is the space of all germs of C∞ mappings Rn → Rp , let J(φ) denote the ideal in generated by φ1, … , φp and the Jacobian determinants

where Let

Clearly, is an ideal in and where is the (unique) maximal ideal of .