Volume 8 (2008), Number 3. Abstracts

A. Bassa, A. Garcia, and H. Stichtenoth. A New Tower over Cubic Finite Fields [PDF]

We present a new explicit tower of function fields (Fn)n≥0 over the finite field with l=q3 elements, where the limit of the ratios (number of rational places of Fn)/(genus of Fn) is bigger or equal to 2(q2−1)/(q+2). This tower contains as a subtower the tower which was introduced by Bezerra–Garcia–Stichtenoth, and in the particular case q=2 it coincides with the tower of van der Geer–van der Vlugt. Many features of the new tower are very similar to those of the optimal wild tower in an earlier work by the second and the third author over the quadratic field Fq2 (whose modularity was shown by Elkies).

Keywords. Towers of function fields, genus, rational places, limits of towers, Zink's bound, cubic finite fields, Artin–Schreier extensions.

2000 Mathematics Subject Classification. 11R58, 11G20, 14G35, 14G05.

C. Boldrighini, R. Minlos, F. Nardi, and A. Pellegrinotti. Asymptotic Decay of Correlations for a Random Walk on the Lattice Zd in Interaction with a Markov Field [PDF]

We consider a discrete-time random walk on Zd, d=1, 2, … in a random environment with Markov evolution in time. We complete and extend to all dimension d≥1 our earlier results on the time decay of the correlations of the “environment from the point of view of the random walk”.

Keywords. Random walks, Markov chains, correlations, environment from the point of view.

2000 Mathematics Subject Classification. 60J15, 82B10, 82B41.

A. Esterov. On the Existence of Mixed Fiber Bodies [PDF]

We give a direct proof of the existence of mixed fiber bodies which is not based upon the reduction to polytopes by continuity. It uses an explicit formula for the support function of a mixed fiber body instead. With the aid of this formula one can also compute the support function of a Newton polytope of a multidimensional resultant and prove a certain monotonicity property for mixed fiber bodies.

Keywords. Fiber polytope, mixed volume, convexity, resultant.

2000 Mathematics Subject Classification. 52A20, 52A39, 52B20.

A. Hamilton and A. Lazarev. Symplectic C-algebras [PDF]

In this paper we show that a strongly homotopy commutative (or C-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C-algebra (an ∞-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a C-algebra and does not generalize to algebras over other operads.

Keywords. Infinity-algebra, cyclic cohomology, Harrison cohomology, symplectic structure, Hodge decomposition.

2000 Mathematics Subject Classification. 13D03, 13D10, 46L87, 55P62.

V. Kleptsyn and A. Navas. A Denjoy Type Theorem for Commuting Circle Diffeomorphisms with Derivatives Having Different Hölder Differentiability Classes [PDF]

Let d≥2 be an integer number, and let fk, k∈{1,…,d}, be C1+τk commuting circle diffeomorphisms, with τk∈]0,1[ and τ1+…+τd>1. We prove that if the rotation numbers of the fk's are independent over the rationals (that is, if the corresponding action of Zn on the circle is free), then they are simultaneously (topologically) conjugate to rotations.

Keywords. Denjoy theorem, centralizers, Hölder class of the derivative.

2000 Mathematics Subject Classification. 22F05, 37C15, 37C80, 37E10.

I. Losev. Combinatorial Invariants of Algebraic Hamiltonian Actions [PDF]

To any Hamiltonian action of a reductive algebraic group G on a smooth irreducible symplectic variety X we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. For cotangent bundles these invariants essentially coincide with those arising in the theory of equivariant embeddings. Using our approach we establish some properties of the latter invariants.

Keywords. Reductive groups, Hamiltonian actions, cotangent bundles, Weyl groups, root lattices.

2000 Mathematics Subject Classification. 14L30, 53D20.

L. Ortiz-Bobadilla, E. Rosales-González, and S. Voronin. Analytic Normal Forms of Germs of Holomorphic Dicritic Foliations [PDF]

We consider the class Vn+1d of dicritic germs of holomorphic vector fields in (C2,0) with vanishing n-jet at the origin, n≥1, and their generated foliations. Earlier we proved that under some genericity assumptions, the formal equivalence of two germs implies their analytic equivalence and formal normal forms of germs in Vn+1d were given. In this work we give analytic normal forms of generic germs of dicritic foliations of Vn+1d.

Keywords. Dicritic foliations, dicritic vector fields, normal forms, formal normal forms, analytic normal forms, formal equivalence, analytical equivalence.

2000 Mathematics Subject Classification. 34M35, 37F75.

C. Rousseau and L. Teyssier. Analytical Moduli for Unfoldings of Saddle-Node Vector Fields [PDF]

In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic classification under orbital equivalence and a complete modulus of analytic classification under conjugacy. The modulus is an unfolding of the corresponding modulus for the germ of a vector field with a saddle-node. The point of view is to compare the family with a “model family” via an equivalence (conjugacy) over canonical sectors. This is done by studying the asymptotic homology of the leaves and its consequences for solutions of the cohomological equation.

Keywords. Holomorphic foliation, analytical classification, unfolding of singularities.

2000 Mathematics Subject Classification. 34A20, 34A26, 34C20.

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Moscow Mathematical Journal
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© 2008, Independent University of Moscow

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