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This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for a submodule of finite length of a free module over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of is not equal to , is the multiplicity of .

Specifically, we apply the multiplicity polar theorem to the study of stratification conditions on families of hypersurfaces, obtaining the first set of invariants giving necessary and sufficient conditions for the A condition for hypersurfaces with non-isolated singularities.

These are notes of the introductory courses on the subject we lectured in Trieste in 2003 and Luminy in 2004. The lectures contain basic notions and fundamental theorems of the local theory.

In deformations of polynomial functions one may encounter “singularity exchange at infinity” when singular points disappear from the space and produce “virtual” singularities which have an influence on the topology of the limit polynomial. We find several rules of this exchange phenomenon, in which the total quantity of singularity turns out to be not conserved in general.

A classical theorem of Briançon, Speder and Teissier states that a family of isolated hypersurface singularities is Whitney equisingular if, and only if, the *-sequence for a hypersurface is constant in the family. This paper shows that the constancy of relative polar multiplicities and the Euler characteristic of the Milnor fibres of certain families of non-isolated singularities is equivalent to the Whitney equisingularity of a family of corank 1 maps from -space to + 1-space. The number of invariants needed is 4 − 2, which greatly improves previous general estimates.

Consider a periodic function of two variables with symmetry Γ and let ⊂ Γ be the subgroup of translations. The Fourier expansion of a periodic function is a sum over *, the dual of the set of all the periods of . After projecting , some of its original symmetry remains. We describe the symmetries of the projected function, starting from Γ and from the structure of *.

Unimodular functions have a -constant line in their miniversal unfoldings. Their miniversal deformations on the other hand contain a nontrivial -constant stratum only for the three cases of elliptic singularities. In computer experiments we found six sub-series of the -series, which have a modular line in the their miniversal deformations. The singular locus of the family restricted to such a line splits into an elliptic singularity and another one of -type, such that the deformation is -constant along the modular line. Each modular line can be patched together with the modular line of the associated elliptic singularity, completing it at infinity. All computations are based on the author’s algorithm for computing modular spaces as flatness stratum of the relative cotangent cohomology inside a deformation.

It is known that Goursat distributions (subbundles in the tangent bundles having the tower of consecutive Lie squares growing in ranks very slowly, always by one) possess, from corank 8 onwards, numerical moduli of the local classification, in both C and real analytic categories. (Whereas up to corank 7 that classification is discrete, as shown in a series of papers, the last in that series being [].)

A natural question, first asked by A.Agrachev in 2000, is whether the moduli of Goursat distributions descend to the level of nilpotent approximations: whether they are stiff enough to survive the passing to the nilpotent level. In the present work we show that it is not the case for the first modulus appearing in corank 8 (and the only one known to-date in that corank).

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension in terms of differential forms. In the case = 1 such computations have many applications in differential equations and counting their limit cycles. For > 3, these computations give us an explicit definition of Hodge cycles.

We study how to minimize the number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of any finitely determined holomorphic germ : (ℂ, 0) → (ℂ, 0), with > 3. Gaffney showed in [] that the invariants for the Whitney equisingularity are the 0-stable invariants and the polar multiplicities of the stable types of the germ. First we describe all stable types which appear in these dimensions. Then we find relationships between the polar multiplicities of the stable types in the singular set and also in the discriminant. When > 3, for any germ there is an hypersurface in ℂ, which is of special interest, the closure of the inverse image of the discriminant by , which possibly is with non isolated singularities. For this hypersurface we apply results of Gaffney and Gassler [], and Gaffney and Massey [], to show how the Lê numbers control the polar invariants of the strata in this hypersurface. Gaffney shows that the number of invariants needed is 4+10. In the corank one case we reduce this number to 2+2. The polar multiplicities are also an interesting tool to compute the local Euler obstruction of a singular variety, see []. Here we apply this result to obtain explicit algebraic formulae to compute the local Euler obstruction of the stable types which appear in the singular set and also for the stable types which appear in the discriminant, of corank one map germs from ℂ to ℂ with ≥ 3.

Let be a hypersurface of degree in (ℂ) with isolated singularities, and let : ℂ → ℂ be a homogeneous equation for .

The singularities of can be simultaneously versally deformed by deforming the equation , in an affine chart containing all of the singularities, by the addition of all monomials of degree at most , for sufficiently large ; it is known (see, e.g., §1) that ≥(−2) suffices. Conversely, if the addition in the affine chart of all monomials of degree at most (−2)−1−, ≥ 0, fails to simultaneously versally deform the singularities of , then we will say that is .

The first main result of this paper shows that is -non-versal if, and only if, there exists a homogeneous polynomial vector field with coefficients of degree , which annihilates but is not Hamiltonian for .

Our second main result is a sufficient condition for an -non-versal hypersurface to be -versal.