On an open problem of Niederreiter (Q1870025)

The author looks at the problem of Niederreiter posed in 1991: characterize the polynomials in \(F_q[x_1, \dots,x_n]\) which are permutation polynomials over every finite extension of the field \(F_q\), where \(q\) is the power of a prime \(p\). The author solves the problem when \(n=2\) provided that: \(\text{gcd} (\partial f/ \partial x_1,\partial f/ \partial x_2)=1\) and \(\deg f\not \equiv 0\pmod p\).