$ZZ\gamma$ and $Z\gamma \gamma$ couplings in $\gamma e$ collision with polarized beams

The potential of $\gamma$e mode of linear $e^{+}{e}^{-}$ collider to probe $ZZ\gamma$ and $Z\gamma \gamma$ vertices is investigated through the Z boson production from the procees $\gamma e\to Ze$. Considering the longitudinal and transverse polarization states of the Z boson and incoming polarized beams we find the 95% C.L. limits on the form factors $h_3^Z$, $h_4^Z$, $h_3^{\gamma }$ and $h_4^{\gamma }$ with integrated luminosity 500$f{b}^{-1}$ and $\sqrt{s}=$0.5, 1, 1.5 TeV energies. It is shown that the polarization can improve sensitivities by factors 2-3 depending on the energy.