More constructions of permutation pentanomials and hexanomials over \(\mathbb {F}_{p^{2m}}\)

In this paper, two classes of permutation pentanomials over finite fields \(\mathbb {F}_{p^{2m}}\) ${\mathbb{F}}_{p^{2m}}$ are investigated by transforming the permutation property of polynomials to verifying that some low-degree equations has no solutions in the unit circle. Furthermore, based on the study of the algebraic curves for fractional polynomials, several classes of permutation pentanomials and hexanomials over \(\mathbb {F}_{5^{2m}}\) ${\mathbb{F}}_{5^{2m}}$ are discovered. Additionally, we obtain some new permutation pentanomials, quadrinomials and octonomials over \(\mathbb {F}_{5^{2m}}\) ${\mathbb{F}}_{5^{2m}}$ from known permutation polynomials in the unit circle.