Several classes of permutation polynomials of the form $ (x^{p^m}-x+\delta)^s+L(x) $

In this paper, we give six classes of permutation polynomials of the form $ (x^{2^m}+x+\delta)^s+ax $ over $ \mathbb{F}_{2^{2m}} $ and six classes of permutation polynomials of the form $ (x^{p^m}-x+\delta)^s+ax^{p^m}+a^{p^m}x $ over $ \mathbb{F}_{p^{2m}} $ ($ p $ being an odd prime), respectively. In addition, we also investigate permutation polynomials obtained from piecewise functions. Consequently, we find some complete permutation polynomials.