In C n we consider an algebraic surface Y and a finite collection of hypersurfaces {S i }. Froissart's theorem states that if Y and {S i } are in general position in the projective compactification of C n together with the hyperplane at infinity then for the homologies of Y \ S i we have a special decomposition in terms of the homology of Y and all possible intersections of S i in Y . We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of C n in which Y and {S i } are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ S k with the exception of one Y ∩ S i , which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.
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.
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