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L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 45, 4 (1995), 897-927 SIMPLE EXPONENTIAL ESTIMATE FOR THE NUMBER OF REAL ZEROS OF COMPLETE ABELIAN INTEGRALS by D. NOVIKOV(1) and S. YAKOVENKO^^2) 1. ABELIAN INTEGRALS AND POLYNOMIAL ENVELOPES OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS One of the main results of this paper is an upper bound for the total number of real isolated zeros of complete Abelian integrals, exponential in the degree of the form (Theorem 1 below). This result improves a previously obtained in [IY1] double exponential estimate for the number of real isolated zeros on a positive distance from the singular locus. In fact, the theorem on zeros of Abelian integrals is a particular case of a more general result concerning the number of zeros in polynomial envelopes of irreducible and essentially irreducible differential operators and equations (see §1.3 below). The first announcement of these and other results proved below was in [NY]. In §1 all principal results are formulated and all necessary definitions gathered, §2 explains connections between Abelian integrals and polynomial envelopes: since the most part of preparatory work was already ll; The research was partially supported by the Minerva Foundation.

We derive an explicit system of Picard Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants appears only in dimension approximately two times greater than the standard Picard Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were able to majorize the number of all (real and complex) zeros. In the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and non-homogeneity of uni-and bivariate complex polynomials are studied. 2001Elsevier Science

Within Abell galaxy clusters containing wide‐angle tailed (WAT) radio sources, there is evidence of a `prevailing wind' which directs the WAT jets. We study the alignment of WAT jets and nearby clusters to test the idea that this wind may be a fossil of drainage along large‐scale supercluster axes. We also test this idea with a study of the alignment of WAT jets and supercluster axes. Statistical tests indicate no alignment of WAT jets towards nearest neighbour clusters, but do indicate approximately 98 per cent confidence in alignment with the long axis of the supercluster in which the cluster lies. We find a preferred scale for such superclusters of order 25 h‐1 Mpc.

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R n and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated.