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We obtain a sufficient solvability condition for Cauchy problems for a polynomial difference operator with constant coefficients. We prove that if the generating function of the Cauchy data of a homogeneous Cauchy problem lies in one of the classes of Stanley's hierarchy then the generating function of the solution belongs to the same class.

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Using the notion of fundamental solution, we obtain a solution to the Cauchy problem for a multidimensional homogeneous linear difference equation with constant coefficients.Denote by Z the set of integers and by Z n = Z × · · · × Z the n-dimensional integer-valued lattice. Let Z n + be the subset of Z n of the points with nonnegative integer coordinates, and fix some finite subsetwhere c α are the (constant) coefficients of the equation.The characteristic polynomial of (1) is the polynomial α∈A c α z α =:The characteristic set of (1) is the set V = {z ∈ C n : P (z) = 0} of zeros of the characteristic polynomial.It is well known (for example, see [1]) that for n = 1 every solution to (1) is a linear combination of solutions of the form x s λ x , s = 0, . . . , k − 1, where λ ∈ V is a root of multiplicity k. Consequently, the solution space of (1) is finite-dimensional and its dimension equals the degree m of P (the order of the equation).Moreover, it is obvious that every solution to (1) is determined completely by the values f (x) = φ x at the "initial" points x = 0, 1, . . . , m − 1.For n > 1 the situation is essentially more complicated, since the solution space is infinite-dimensional and the question about the set on which the "initial" values of solutions should be given is not so obvious (neither the question of definition of the order of the difference equation). Observe that multidimensional difference equations (recurrent relations) arise in combinatorial analysis [2] and in discretization of differential equations (for example, see [3]).In [4] the author gave a description of the solution space of (1) using the notions of the Newton polyhedron and amoeba of the characteristic polynomial. In this article we define the set Z n m on which the "initial" values are given, state the Cauchy problem for (1), and using the notion of fundamental solution solve this problem.The Newton polyhedron N P of the characteristic polynomial P (z) is the convex hull in R n of the elements of A.Fix m ∈ N P ∩ Z n and denote Z n m j = {y ∈ Z n : 0 ≤ y j < m j }. Now, put n j=1 Z n m j =: Z n m , and let φ : Z n m → C be a given function.Problem. Find a solution to (1) coincident with φ on Z n m :f (x) = φ(x), x ∈ Z n m .(2)

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