Volume 7 (2007), Number 2. Abstracts

V. Arnold. Arithmetical Turbulence of Selfsimilar Fluctuations Statistics of Large Frobenius Numbers of Additive Semigroups of Integers [PDF]

The Frobenius number of vector a, whose components a_s are natural numbers (having no common divisor greater than 1), is the minimal integer N(a) which is representable as a sum of the components a_s with nonnegative multiplicities, together with all the greater integers (like for N(4,5)=12).

The mean Frobenius number is the arithmetical mean value of the number N(a) along the simplex of the vectors a for which a₁+...+a_n=σ.

Numerical experiments suggest the growth rate of this mean value (for large σ) of order σ^p, where p=1+1/(n-1), that is of order 3/2 for N(a,b,c).

Fluctuations are making some of the Frobenius numbers many times higher at the place of some resonances, like b=c.

The selfsimilar statistics of the fluctuations, contained in the present article, suggest, that these fluctuations are insufficiently frequent to influence the behaviour of the mean value at large scales σ.

Keywords. Fluctuation, statistics, weak asymptotics, Diophantine problems, continued fractions, tails, averaging, mean values, growth rate, resonances, scales, selfsimilarity

2000 Mathematics Subject Classification. 11D04, 20M99

V. Batyrev and B. Nill. Multiples of Lattice Polytopes without Interior Lattice Points [PDF]

Let Δ be an n-dimensional lattice polytope. The smallest non-negative integer i such that kΔ contains no interior lattice points for 1≤k≤n-i we call the degree of Δ. We consider lattice polytopes of fixed degree d and arbitrary dimension n. Our main result is a complete classification of n-dimensional lattice polytopes of degree d=1. This is a generalization of the classification of lattice polygons (n=2) without interior lattice points due to Arkinstall, Khovanskii, Koelman and Schicho. Our classification shows that the secondary polytope Sec(Δ) of a lattice polytope of degree 1 is always a simple polytope.

Keywords. Lattice polytope, principal A-determinant

2000 Mathematics Subject Classification. Primary: 52B20; Secondary: 14M25

A. Belov-Kanel and M. Kontsevich. The Jacobian Conjecture is Stably Equivalent to the Dixmier Conjecture [PDF]

The paper is devoted to the proof of equivalence of Jacobian and Dixmier conjectures. We show that 2n-dimensional Jacobian conjecture implies Dixmier conjecture for W_n. The proof uses “antiquantization”: positive characteristics and Poisson brackets on the center of Weyl algebra in characteristic p.

Keywords. Poisson brackets, symplectic structure, quantization, polynomial automorphism, Weyl algebra, differential operator, Jacobian conjecture

2000 Mathematics Subject Classification. 16S32, 16S80, 14R15

V. Buchstaber, T. Panov, and N. Ray. Spaces of Polytopes and Cobordism of Quasitoric Manifolds [PDF]

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details.

Keywords. Analogous polytopes, complex cobordism, connected sum, framing, omniorientation, quasitoric manifold, stable tangent bundle

2000 Mathematics Subject Classification. 55N22, 52B20, 14M25

A. Campillo, F. Delgado, and S. M. Gusein-Zade. On Poincaré Series of Filtrations on Equivariant Functions of Two Variables [PDF]

Let a finite group G act on the complex plane (C²,0). We consider multi-index filtrations on the spaces of germs of holomorphic functions of two variables equivariant with respect to 1-dimensional representations of the group G defined by components of the exceptional divisor of a modification of the complex plane C² at the origin or by branches of a G-invariant plane curve singularity (C,0)⊂(C²,0). We give formulae for the Poincaré series of these filtrations. In particular, this gives a new method to obtain the Poincaré series of analogous filtrations on the rings of germs of functions on quotient surface singularities.

Keywords. Equivariant functions, filtrations, Poincaré series

2000 Mathematics Subject Classification. 14B05, 16W70, 16W22

A. Degtyarev, I. Itenberg, and V. Kharlamov. Deformation Finiteness for Real Hyperkähler Manifolds [PDF]

We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperkähler manifolds is finite.

Keywords. Hyperkähler manifold, Calabi–Yau families, real structure, equivariant deformation, finiteness results

2000 Mathematics Subject Classification. 53C26, 14P25, 14J32, 32Q20

D. Fuchs and E. Fuchs. Closed Geodesics on Regular Polyhedra [PDF]

We give a description of closed geodesics, both self-intersecting and non-self-intersecting, on regular tetrahedra, cubes, octahedra and icosahedra.

Keywords. Regular polyhedra, closed geodesics, simple geodesics, prime geodesics

2000 Mathematics Subject Classification. 53C22

A. Glutsyuk and Yu. Ilyashenko. Restricted Version of the Infinitesimal Hilbert 16th Problem [PDF]

The paper deals with an abelian integral of a polynomial 1-form along a family of real ovals of a polynomial (hamiltonian) in two variables (the integral is considered as a function of value of the Hamiltonian). We give an explicit upper bound on the number of its zeroes (assuming the Hamiltonian ultra-Morse of arbitrary degree and ranging in a compact subset in the space of ultra-Morse polynomials of a given degree, and that the form has smaller degree). This bound depends on the choice of the compact set and is exponential in the fourth power of the degree.

Keywords. Two-dimensional polynomial Hamiltonian vector field, oval, polynomial 1-form, Abelian integral, complex level curve, critical value, vanishing cycle

2000 Mathematics Subject Classification. Primary: 58F21; 14K20, Secondary: 34C05

B. Sturmfels, J. Tevelev, and J. Yu. The Newton Polytope of the Implicit Equation [PDF]

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.

Keywords. Implicitization, Newton polytope, tropical geometry

2000 Mathematics Subject Classification. 13P10, 14Q99, 52B20, 68W30

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