Multidimensional versions of Poincaré's theorem for difference equations

On the Asymptotic of Homological Solutions to Linear Multidimensional Difference Equations

“…Theorem (Leinartas, Passare, Tsikh [6]). If for the direction q ∈ QP n−1 the roots λ (j) (q) of the limit characteristic system (8) are such that the absolute values of all monomials [λ (j) (q)] q are different, then for any solution f (x) of the form (7) non-vanishing on the sequence {a + lq}, a ∈ Z n , the limit of the Horn vector…”

“…In [6] we investigated the connection between combinations (7) of fundamental solutions (6) and homological solutions (4) in the case k = n − 1. In section 3 we shall complete the list of fundamental solutions (see formula (17)) and describe their connection with homological solutions (4) for k n − 1 (see Proposition 1).…”

“…Like homological solutions (4) these fundamental solutions are defined by integrals, but the integration cycles here lie outside the characteristic set. Namely, in [6] to each connected component E ν of the amoeba complement R n \A V a fundamental solution is associated by means of the integral…”

“…for half-dimensional cycles γ. For such solutions a multidimensional analog of Poincaré's theorem was proved in [6], where instead of the ratio f (x + 1)/f (x) the authors considered the vector (see section 2)…”

“…In the paper [6] solutions (4) were considered only for k = n − 1, i.e. for half-dimensional cycles γ.…”