Residue currents of the Bochner-Martinelli type

Our objective is to construct residue currents from Bochner-Martinelli type kernels; the computations hold in the non complete intersection case and provide a new and more direct approach of the residue of Coleff-Herrera in the complete intersection case; computations involve crucial relations with toroidal varieties and multivariate integrals of the Mellin-Barnes type

Fourier Quasicrystals on Rn\mathbb R^n Preliminary Report

This paper has three aims. First, for $n \geq 1$ we construct a family of real-rooted trigonometric polynomial maps $P : \mathbb C^n \mapsto \mathbb C^n$ whose divisors are Fourier Quasicrystals (FQ). For $n = 1$ these divisors include the first nontrivial FQ with positive integer coefficients constructed by Kurasov and Sarnak \cite{kurasovsarnak}, and for $n > 1$ they overlap with Meyer's curved model sets \cite{meyer6} and two-dimensional \cite{meyer7} and multidimensional \cite{meyer8} crystalline measures. We prove that the divisors are FQ by directly computing their Fourier transforms using a formula derived in \cite{lawton}. Second, we extend the relationship between real-rootedness and amoebas, derived for $n = 1$ by Alon, Cohen and Vinzant \cite{alon}, to the case $n > 1.$ The extension uses results in \cite{bushuevatsikh} about homology of complements of amoebas of algebraic sets of codimension $> 1.$ Third, we prove that the divisors of all uniformly generic real-rooted $P$ are FQ. The proof uses the formula relating Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii \cite{gelfondkhovanskii1} . Finally, we note that Olevskii and Ulanovskii [60] have proved that all FQ are divisors of real-rooted trigonometric polynomials for $n = 1$ but that the situation for $n > 1$ remains unsolved

Multidimensional Fourier Quasicrystals I. Sufficient Conditions

We derive sufficient conditions for an atomic measure $\sum_{\lambda \in \Lambda} m_\lambda\, \delta_\lambda,$ where $\Lambda \subset \mathbb R^n,$ $m_\lambda$ are positive integers, and $\delta_\lambda$ is the point measure at $\lambda,$ to be a Fourier quasicrystal, and suggest why they may also be necessary. These conditions extend the necessary and sufficient conditions derived by Lev, Olevskii, and Ulanovskii for $n = 1.$ Our methods exploit the toric geometry relation between Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii.Comment: 17 page

Amoebas of complex hypersurfaces in statistical thermodynamics

The amoeba of a complex hypersurface is its image under a logarithmic projection. A number of properties of algebraic hypersurface amoebas are carried over to the case of transcendental hypersurfaces. We demonstrate the potential that amoebas can bring into statistical physics by considering the problem of energy distribution in a quantum thermodynamic ensemble. The spectrum ${\epsilon_k}\subset \mathbb{Z}^n$ of the ensemble is assumed to be multidimensional; this leads us to the notions of a multidimensional temperature and a vector of differential thermodynamic forms. Strictly speaking, in the paper we develop the multidimensional Darwin and Fowler method and give the description of the domain of admissible average values of energy for which the thermodynamic limit exists.Comment: 18 pages, 5 figure

Domains of convergence for Ahypergeometric series and integrals

We prove two theorems on the domains of convergence for A-hypergeometric series and for associated Mellin-Barnes type integrals. The exact convergence domains are described in terms of amoebas and coamoebas of the corresponding principal A-determinant