Volume 7 (2007), Number 3. Abstracts

N. Berline and M. Vergne. Local Euler–Maclaurin Formula for Polytopes [PDF]

We prove a local Euler–Maclaurin formula for rational convex polytopes in a rational Euclidean space. For every affine rational polyhedral cone c in V, we construct a differential operator of infinite order D(c) on V with constant rational coefficients. Then for every convex rational polytope p in V and every polynomial function h(x) on V, the sum of the values of h(x) at the integral points of p is equal to the sum, for all faces f of p, of the integral over f of the function D(t(p,f))⋅h where we denote by t(p,f) the transverse cone of p along f, an affine cone of dimension equal to the codimension of f. Applications to numerical computations when p is a polygon are given.

Keywords. Lattice polytope, valuation, Euler–Maclaurin formula, toric varieties

2000 Mathematics Subject Classification. 52

F. Bihan and F. Sottile. New Fewnomial Upper Bounds from Gale Dual Polynomial Systems [PDF]

We show that there are fewer than ((e2+3)/4) 2\binom{k}{2} nk positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This is significantly smaller than Khovanskii's fewnomial bound of 2\binom{n+k}{2}(n+1)n+k. We reduce the original system to a system of k equations in k variables which depends upon the vector configuration Gale dual to the exponents of the monomials in the original system. We then bound the number of solutions to this Gale system. We adapt these methods to show that a hypersurface in the positive orthant of Rn defined by a polynomial with n+k+1 monomials has at most C(k)nk−1 compact connected components. Our results hold for polynomials with real exponents.

Keywords. Fewnomials, Gale dual, sparse polynomial

2000 Mathematics Subject Classification. 14P99

S. Chmutov and I. Pak. The Kauffman Bracket of Virtual Links and the Bollobás–Riordan Polynomial [PDF]

We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás–Riordan polynomial RGL of a ribbon graph associated with L. This result generalizes the celebrated relation between the classical Kauffman bracket and the Tutte polynomial of planar graphs.

Keywords. Virtual knots and links, knot invariants, Jones polynomial, Kauffman bracket, Tutte polynomial, Bollobás–Riordan polynomial, ribbon graph

2000 Mathematics Subject Classification. 57M15, 57M27, 05C10, 05C22

A. Costa and S. Natanzon. Classification of Zpkm Orientation Preserving Actions on Surfaces [PDF]

In this article we completely classify orientation preserving actions of groups Zpkm (p is a prime integer) on compact oriented surfaces.

Keywords. Action, surface, automorphisms

2000 Mathematics Subject Classification. 57M12, 30F10

A. Dickenstein, J. M. Rojas, K. Rusek, and J. Shih. Extremal Real Algebraic Geometry and A-Discriminants [PDF]

We present a new, far simpler family of counterexamples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the A-discriminant, and give new bounds on the topology of certain A-discriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations.

Keywords. Sparse polynomial, real root, discriminant, isotopy, maximal, explicit bound

2000 Mathematics Subject Classification. Primary: 14P25; Secondary: 14M25, 34C08

A. Gabrielov. Counterexamples to Quantifier Elimination for Fewnomial and Exponential Expressions [PDF]

We construct a family of semialgebraic sets of bounded fewnomial complexity, with unbounded fewnomial complexity of their projections to a subspace. This implies impossibility of fewnomial quantifier elimination. We also construct a set defined by exponential algebraic functions such that its projection cannot be defined by a quantifier-free formula with exponential algebraic functions, even if division is permitted. Similar examples are constructed for the unrestricted frontier of fewnomial and exponential semialgebraic sets, and for the Hausdorff limits of families of such sets.

Keywords. Fewnomials, quantifier elimination

2000 Mathematics Subject Classification. Primary: 14P10; Secondary: 14P15

V. Ivrii. Magnetic Schrödinger Operator: Geometry, Classical and Quantum Dynamics and Spectral Asymptotics [PDF]

I study the Schrödinger operator with the strong magnetic field, considering links between geometry of magnetic field, classical and quantum dynamics associated with operator and spectral asymptotics. In particular, I will discuss the role of short periodic trajectories.

Keywords. Magnetic Schrödinger operator, dynamics, periodic trajectories logarithmic uncertainty principle

2000 Mathematics Subject Classification. 35P20

M. Jibladze and D. Novikov. Unimodularity of Poincaré Polynomials of Lie Algebras for Semisimple Singularities [PDF]

We single out a large class of semisimple singularities with the property that all roots of the Poincaré polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra lie on the unit circle; for a still larger class there might occur exactly four roots outside the unit circle. This is a corrected version of a theorem by Elashvili and Khimshiashvili.

Keywords. Isolated semisimple singularities, moduli algebra, derivations, Poincaré polynomial, palindromic polynomials

2000 Mathematics Subject Classification. 32S10

V. Kiritchenko. On Intersection Indices of Subvarieties in Reductive Groups [PDF]

In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined earlier by the author) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups thus extending the beautiful result by D. Bernstein and Khovanskii, which holds for a complex torus. Second, for any regular compactification of a reductive group, it computes the intersection indices of the Chern classes of the compactification with hypersurfaces. The formula is similar to the Brion–Kazarnovskii formula for the intersection indices of hypersurfaces in reductive groups. The proof uses an algorithm of De Concini and Procesi for computing such intersection indices. In particular, it is shown that this algorithm produces the Brion–Kazarnovskii formula.

Keywords. Reductive groups, Chern classes, Euler characteristic of hyperplane sections

2000 Mathematics Subject Classification. 14L30

G. Mikhalkin and A. Okounkov. Geometry of Planar Log-Fronts [PDF]

The log-front of two curves P and Q in a toric surface is the set of torus elements τ such that τ⋅Q is tangent to P. Log-fronts generalize dual curves, wave fronts, and arise naturally in the theory of random surfaces. Our goal in this paper is to prove analogs of Plücker and Klein formulas for log-fronts.

Keywords. Log-front, frozen boundary, Plücker formula, Klein formula

2000 Mathematics Subject Classification. 14P99

E. Mukhin, V. Tarasov, and A. Varchenko. Higher Lamé Equations and Critical Points of Master Functions [PDF]

Under certain conditions, we give an estimate from above on the number of differential equations of order r+1 with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of solutions. The estimate is given in terms of a suitable weight subspace of the tensor power U(n)⊗(n−1), where n is the number of singular points in C and U(n) is the enveloping algebra of the nilpotent subalgebra of glr+1.

Keywords. Lame equation, master function, critical points, quasi-polynomial flag of solutions

2000 Mathematics Subject Classification. 34M35 (17B10 33C05 82B20 82B23)

A. Pukhlikov. Explicit Examples of Birationally Rigid Fano Varieties [PDF]

We construct explicit examples of divisorially canonical Fano double spaces. Their direct products are birationally rigid Fano varieties with finitely many structures of a rationally connected fiber space.

Keywords. Birational geometry, birational rigidity, rationally connected fiber space, linear system, log canonical singularity

2000 Mathematics Subject Classification. 14E05

V. Timorin. Rectifiable Pencils of Conics [PDF]

We describe analytic pencils of conics passing through the origin in C2 that can be mapped to straight lines locally near the origin by an analytic diffeomorphism. Under a minor non-degeneracy assumption, we prove that in a pencil with this property, almost all conics have 3 points of tangency with the same algebraic curve of class 3 (i.e. a curve projectively dual to a cubic).

Keywords. Rectifiable pencil of conics, fractional quadratic map

2000 Mathematics Subject Classification. 51N15

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