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 $(\gamma,\omega)$, where $\gamma$ is a homological $k$-dimensional cycle on the characteristic set of the equation and $\omega$ is a holomorphic form of degree $k$. This pair defines a so called homological solution by the integral over $\gamma$ of the form $\omega$ 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.