Further results on permutation polynomials from trace functions

For a prime p and positive integers m, n, let ${{\mathbb {F}}}_q be a finite field with q=p^m elements and {{\mathbb {F}}}_{q^n} be an extension of {{\mathbb {F}}}_q. Let h(x) be a polynomial over {{\mathbb {F}}}_{q^n} satisfying the following conditions: (i) {\mathrm{Tr}}_m^{nm}(x)\circ h(x)=\tau (x)\circ {\mathrm{Tr}}_m^{nm}(x) ; (ii) For any s \in {{\mathbb {F}}}_{q} , h(x) is injective on...