Polynomial Form Factors in the O(3) Nonlinear sigma-Model

We study the general structure of Smirnov's axioms on form factors of local operators in integrable models. We find various consistency conditions that the form factor functions have to satisfy. For the special case of the $O(3)$ $\sigma$-model we construct simple polynomial solutions for the operators of the spin-field, current, energy-momentum tensor and topological charge density.