Abstract
We consider difference equations of order k
n+k
≥ 2 of the form: y
n+k
= f(yn,...,yn+k-1), n= 0,1,2,... where f: D
k
→ D is a continuous function, and D⊆R. We develop a necessary and sufficient condition for the existence of a symmetric invariant I(x
1
,...,x
k
) ∈C
∞
[D
k
,D]. This condition will be used to construct invariants for linear and rational difference equations. Also, we investigate the transformation of invariants under invertible maps. We generalize and extend several results that have been obtained recently.