Natural Operators Lifting Vector Fields to Bundles of Weil Contact Elements

Let A be a Weil algebra. The bijection between all natural operators lifting vector fields from m-manifolds to the bundle functor K ^A of Weil contact elements and the subalgebra of fixed elements SA of the Weil algebra A is determined and the bijection between all natural affinors on K ^A and SA is deduced. Furthermore, the rigidity of the functor K ^A is proved. Requisite results about the structure of SA are obtained by a purely algebraic approach, namely the existence of nontrivial SA is discussed.     

The interlace polynomial of a graph

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable “interlace polynomial” for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial.It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.     

Fractional Power Series and Pairings on Drinfeld Modules

Let be an algebraically closed field containing which is complete with respect to an absolute value . We prove that under suitable constraints on the coefficients, the series converges to a surjective, open, continuous -linear homomorphism whose kernel is locally compact. We characterize the locally compact sub--vector spaces of which occur as kernels of such series, and describe the extent to which determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of of a given valuation, given the valuations of the coefficients. The ``adjoint' series converges everywhere if and only if does, and in this case there is a natural bilinear pairing

which exhibits as the Pontryagin dual of . Many of these results extend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module.

     

The Bertini irreducibility theorem for higher codimensional slices

In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.     

Irreducibility and the distribution of some exponential sums

We relate the distribution of the absolute value of some generalized Gauss sums to the absolute irreducibility of some polynomials in two variables in characteristic 0 and p.     

Islands of stability of the d-wave order parameter in s-wave anisotropic superconductors

In this paper we find and present on diagrams in the coordinates of η=2t₁/t₀ (the ratio of the second and the first nearest neighbor hopping integrals) and n (the carrier concentration) the areas of stability for the superconducting spin-singlet s- and d-wave and the spin-triplet p-wave order parameters hatching out during the phase transition from the normal to the superconducting phase. The diagrams are obtained for an anisotropic two-dimensional superconducting system with a relatively wide partially-filled conduction band. We study a tight-binding model with an attractive nearest neighbor interaction with the amplitude V₁, and the on-site interaction (with the amplitude V₀) taken either as repulsive or attractive. The problem of the coexistence of the s-, p- and d-wave order parameters is addressed and solved for chosen values of the ratio V₀/V₁. A possible island of stability of the d-wave order parameter in the s-wave order parameter environment for a relatively strong on-site interaction is revealed. The triple points, around which the s-, d-, and p-wave order parameters coexist, are localized on diagrams. It is shown that results of the calculations performed for the two-dimensional tight-binding band model are dissimilar with some obtained within the BCS-type approximation.     

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields . The Weil estimate of such sums is well known: , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, , for which attains the largest possible value and, in particular, .