Adam Parusinski

Proof of the gradient conjecture of R. Thom

Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t )a tx0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x0 onto the unit sphere has flnite length.

Algebraically constructible functions

Abstract An algebraic version of Kashiwara and Schapiras calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and Kings numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8 and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(Xω), where Xω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(Xω), where Xω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization.

Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (ℂ2, 0)→(ℂ, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families ft: (ℂ2, 0)→(ℂ, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.

A note on singularities at infinity of complex polynomials

Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family f of projective closures of fibres of f . We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the C∞-triviality of f . If the support of sheaf of vanishing cycles of f is a finite set, then it detects precisely the change of the topology of the fibres of f . Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres. Let f : C → C be a polynomial function. A value t0 ∈ C of f is called typical if f is a C∞-trivial fibration over a neighbourhood of t0 and atypical otherwise. The set of atypical values, called the bifurcation set of f , consists of the critical values of f and, maybe, some other values coming from the “singularities of f at infinity”. What is “the singularity at infinity” is understood rather heuristically and, in general, no precise definition exists. Under various assumptions this notion can be given a precise meaning, for instance if “the singularities at infinity” are in some sense isolated as in [Pa], [S-T], or [Z]. Consider the family f : X → C of projective closures of fibres of f , X being the closure of the graph of f in P ×C. In this paper we study the singularities of f from the point of view of vanishing cycles of f . In particular, as we show below (Theorem 1.2), the absence of vanishing cycles of f − t0 guarantees that t0 is typical. Thus we 1991 Mathematics Subject Classification: Primary 32S15; Secondary 32S25, 32S55 The paper was written during the author’s stay at Laboratoire Jean-Alexandre Dieudonne in Nice. He would like to express his gratitude to his colleagues at the Laboratoire, in particular to Michel Merle, for a friendly atmosphere and for many helpful discussions concerning the paper. The author also would like to thank Claude Sabbah and Mihai Tibăr for remarks and discussions related to the paper. The paper is in final form and no version of it will be published elsewhere.

Topological Triviality of μ-Constant Deformations of Type f(x) + tg(x)

We show that every μ-constant family of isolated hypersurface singularities of type F ( x , t ) = f ( x )+ tg ( x ), where t is a parameter, is topologically trivial. The proof uses only the curve selection lemma, and hence, for an appropriately translated statement, also works over the reals and for some families of non-isolated singularities. Some applications to study the singularities at infinity of complex polynomials are given.

On the Euler characteristic of fibres of real polynomial maps

Let Y be a real algebraic subset of R and F : Y → R be a polynomial map. We show that there exist real polynomial functions g1, . . . , gs on R such that the Euler characteristic of fibres of F is the sum of signs of gi. The purpose of this paper is to give a new, self-contained and elementary proof of the following result. Theorem. Let Y be a real algebraic subset of R and F : Y → R be a polynomial map. Then there exist real polynomials g1(y), . . . , gs(y) on R n such that the Euler characteristic of fibres of F is the sum of signs of gi, that is χ(F−1(y)) = sgn g1(y) + . . .+ sgn gs(y). Our proof is based on a classical and elementary result expressing the number of real roots of a real polynomial of one variable as the signature of an associated quadratic form known already to Hermite [He1, He2] and Sylvester [Syl], see also [B], [BW], [BCR, p. 97]. In the proof we use a modern generalized version of this result presented in [PRS] (note that we need only a one variable case of [PRS], that is precisely [BR, Proposition p. 18]). Our original proof of the theorem [PS] used different means such as the theory of local topological degree of polynomial mappings, Grobner bases and the Eisenbud-Levine Theorem and was not so explicit as the one presented below. 1991 Mathematics Subject Classification: Primary 14P05, 14P25. Received by the editors: January 23, 1997; in the revised form: November 15, 1997. The paper is in final form and no version of it will be published elsewhere.

Newton-Puiseux roots of Jacobian determinants

Let

Blow-analytic equivalence of two variable real analytic function germs

f(x,y), g(x,y)

On the preparation theorem for subanalytic functions

denote either a pair of holomorphic function germs, or a pair of monic polynomials in