Exemples de Lattès et domaines faiblement sphériques de ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 We describe the attracting basins of the origin in ℂ ^k+1 for the polynomial lifts of Lattès examples. We show that the boundary of these bounded pseudoconvex domains is a quotient of a compact spherical hypersurface, and we describe the singularities that appear. These domains are surprising, because they are very close to the ball, and admit non injective proper holomorphic self-maps. We also explicit some Lattès examples in dimension 2. Received: 17 October 2002 / Revised version: 7 February 2003 Published online: 19 May 2003 Mathematics Subject Classification (2000): 14K25, 32S25, 32T99, 32H50     

Feuilletages holomorphes locaux et analyticité partielle d'applications C ∞

 Let f : M → M′ be a smooth CR mapping between a generic real analytic submanifold M ⊂ ℂ ⁿ , n > 1, and a real analytic subset M′ ⊂ ℂ ^n′ . We prove that if M is minimal and if M′ does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M ^′ . Received: 7 August 2001 Mathematics Subject Classification (2000): 32V25, 32V40, 32H99     

Polynomial and regular images of ℝ n

We obtain new necessary conditions for ann-dimensional semialgebraic subset of ℝ ⁿ to be a polynomial image of ℝ ⁿ . Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers. Supported by Spanish GAAR BFM2002-04797 and European RAAG HPRN-CT-2001-00271.     

Preimages of CR submanifolds with generic holomorphic maps

We investigate the notion of CR transversality of a generic holomorphic map f: ℂ ⁿ → ℂ ^m to a smooth CR submanifold M of ℂ ^m . We construct a stratification of the set of non-CR transversal points in the preimage M′ = f ⁻¹ (M) by smooth submanifolds, consisting of points where the CR dimension of M′ is constant. We show the existence of a Whitney stratification for sets which are locally diffeomorphic to the product of an open set and an analytic set. Work on this paper was supported by ARRS, Republic of Slovenia.     

Invariant currents and dynamical Lelong numbers

Let ƒ be a polynomial automorphism of ℂ^k of degree λ, whose rational extension to ℙ^k maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙ^k of bidegree (1,1), we show that the sequence λ⁻ⁿ(ƒⁿ)*S converges in the sense of currents on ℙ^k to a linear combination of the Green current T₊ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for ƒ⁻¹.     

Convolution operators in A ?∞ for convex domains

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ ⁿ and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.     

On the Malgrange condition for complex polynomials of two variables

 We show that if P  ℂ[X, Y] is nonconstant, the set K _∞ (P) of points where the Malgrange condition fails to hold coincides with the roots of a constructible polynomial in one variable. This strengthens weaker related results in the literature. The proof given here is purely analytical and bypasses the algebraic and topological arguments involved in other approaches. Applications to both the real and complex Jacobian Conjectures and to the reducibility of P-z are discussed. Received: 11 October 2001 / Revised version: 2 April 2002 Mathematics Subject Classification (2000): 12D05, 14D06, 14R15, 32B10     

Congruence properties of the coefficients of the Drinfeld modular invariant

 Let be the expansion at infinity of the Drinfeld modular invariant. We know that the coefficients c _n 's are in the polynomial ring 𝔽_q[T]. In this text, we prove for these coefficients congruence properties modulo powers of p, where p  𝔽 _q [T] is a polynomial of degree 1. Received: 19 April 2002 Mathematics Subject Classification (2000): 11F52, 11F33, 11G09     

Rational curves and rational singularities

 We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ^*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation. Received: 7 May 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 14J17, 14L30, 13H10     

Effective algebraic degeneracy

We show that for every smooth projective hypersurface X⊂ℙ ⁿ⁺¹ of degree d and of arbitrary dimension n ≥2, if X is generic, then there exists a proper algebraic subvariety Y ⊊ X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound d\geqslant 2ⁿ⁵d\geqslant 2^{n^{5}} .     

On Levi-flat hypersurfaces with given boundary in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let S ⊂ ℂ ⁿ be a compact connected 2-codimensional submanifold. If n ⩾ 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in ℂ ⁿ of a Levi-flat (2n−1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point. For n = 2 many results are known since the 1980’s and a new result with a very technical hypothesis is announced. Dedicated to Professor LU QiKeng on the occasion of his 80th birthday     

On polynomial collocation for second kind integral equations with fixed singularities of Mellin type

Summary.  We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gau? quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with 𝒪(n ²[log n]²) resp. 𝒪(n ²) operations, where n is the dimension of the polynomial trial space. Received February 18, 2002 / Revised version received May 15, 2002 / Published online October 29, 2002 RID="⋆" ID="⋆" Correspondence to: A. Rathsfeld Mathematics Subject Classification (1991): 65R20     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

Minimal, rigid foliations by curves on ℂℙ n

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ ⁿ for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙ ⁿ . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.?This is done by considering pseudo-groups generated on the unit ball 𝔹 ⁿ ⊆ℂ ⁿ by small perturbations of elements in Diff(ℂ ⁿ ,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C ⁰-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙ ⁿ . Received March 7, 2002 / final version received November 26, 2002?Published online February 7, 2003 Most of this work has been carried out during a visit of the first author to IMPA/RJ and a visit of the second author to the University of Lille 1. We would like to thank these institutes for hospitality and express our gratitude to CNPq-Brazil and CNRS-France for the financial support which made these visits possible. We are also indebted to Paulo Sad, Marcel Nicolau and the referee whose comments helped us to improve on the preliminary version. Finally, the second author has partially conducted this research for the Clay Mathematics Institute.     

Polyhedral end games for polynomial continuation

Bernshtein’s theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (ℂ^*)ⁿ, with ℂ^* = ℂ\{0}. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (ℂ^*)ⁿ. In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with approximations of its solutions. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3.     

Relative cohomology of polynomial mappings

 Let F be a polynomial mapping from ℂ ⁿ to ℂ ^q with n>q. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre F ⁻¹(∞) ``at infinity' and its cohomology. Let us fix a weighted homogeneous degree on with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of F. We introduce the cohomology groups H ^k (F ⁻¹(∞)) of F at infinity. These groups enable us to compute all the other cohomology groups of F. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of H ^n−q (F ⁻¹ (∞)) is a basis of all the groups H ^n−q (F ⁻¹(y)) and also a basis of the (n−q) ^th relative cohomology group of F. Moreover the dimension of H ^n−q (F ⁻¹(∞)) is given by a global Milnor number of F, which only depends on the leading terms of the coordinate functions of F. Received: 12 February 2002 / Revised version: 25 May 2002 Published online: 3 March 2003     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

On regular polynomial endomorphisms of ℂ<Superscript>2</Superscript> without bounded critical orbitswithout bounded critical orbits

We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ²↦ℂ² under which all complete external rays from infinity for f have well defined endpoints.     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.