Toshizumi Fukui

Seeking invariants for blow-analytic equivalence

We introduce some blow-analytic invariants of real analytic function-germsand discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.

Newton filtrations, graded algebras and codimension of non-degenerate ideals

The computation of dimCOn/I, the codimension of an ideal I in the ring On of holomorphic map germs at the origin in C, is one of the main tools used to calculate geometric invariants of singularities. In general, this calculus is done using the theory of standard basis of the ring On/I. For weighted homogeneous map germs f : (C, 0) → (C, 0) it is known that some geometric invariants are given in terms of its weights and degrees. For instance, Milnor and Orlik gave a formula in [13] for the Milnor number of a weighted homogeneous map germ f : (C, 0) → (C, 0) with an isolated singularity at the origin. There are other invariants that are also computed for weighted homogeneous map germs

Motivic invariants of real polynomial functions and their Newton polyhedrons

We propose a computation of real motivic zeta functions for real polynomial functions, using Newton polyhedron. As a consequence we show that the weights are blow-Nash invariants of convenient weighted homogeneous polynomials in three variables.

Isolated singularities of binary differential equations of degree n

We study isolated singularities of binary differential equations of degree n which are totally real. This means that at any regular point, the associated algebraic equation of degree n has exactly n different real roots (this generalizes the so called positive quadratic differential forms when n = 2). We introduce the concept of index for isolated singularities and generalize Poincar´e-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the n-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics D1, D2 and D3.

Tame Nonsmooth Inverse Mapping Theorems

We give several versions of local and global inverse mapping theorems for tame, nonnecessarily smooth, mappings. Here, tame mapping means a continuous mapping which is locally definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity of some principal minors) of the space of Jacobian matrices at smooth points.