Let us consider the difference equation \(w(z+1)=P(w(z))\), where P is a polynomial of degree at least two. The authors prove that any entire solution w of this equation satisfies \(\lim_{x\to -\infty}w(x+iy)=b\) with b such that \(P(b)=b\), uniformly in \(a\leq y\leq c\) for any constants a,c; if w is not constant, then either \(| P'(b)| \geq 1\) or \(P'(b)=1.\) They also show how to construct in every case all the solutions of the equation.