Abstract
We show that an autonomous difference equation, of arbitrary order and with one or more independent variables, can be linearized by a point transformation if and only if it admits a symmetry vector field whose coefficient function is the product of two functions, one of the dependent variable u and one of the independent variables x: X(x, u)=A(x)G(u) partial/partial u . The factor depending on the independent variables, A, is required to satisfy some non-degeneracy conditions. This result is derived using a discrete jet space formalism for partial and ordinary difference equations, analogous to that used for the study of differential equations.