$ZZ\gamma$ 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 $\gamma e\to Z e$. 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$fb^{-1}$ 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.