On the Asymptotic of Homological Solutions to Linear Multidimensional Difference Equations

Given a linear homogeneous multidimensional difference equation with constant coefficients, we choose a pair (γ, ω), where γ is a homological k-dimensional cycle on the characteristic set of the equation and ω is a holomorphic form of degree k. This pair defines a so called homological solution by the integral over γ of the form ω multiplied by an exponential kernel. A multidimensional variant of Perron's theorem in the class of homological solutions is illustrated by an example of the first order equation.