Let \(\mathbb {F}_{q^m}\) be the finite field of order \(q^m\) , where \(m\ge 2\) and q a prime power. Given \(\mathbb {F}_q\) -affine hyperplanes \(A_1,\ldots , A_m\) of \(\mathbb {F}_{q^m}\) in general position, we study the existence of a primitive element \(\alpha \) of \(\mathbb {F}_{q^m}\) , such that \(a\alpha ^2+b\alpha +c\) ( \(a\ne 0\) and \(b^2\ne 4ac\) ) is also primitive in \(\mathbb {F}_{q^m}\) and the primitive pair avoids \(A_i\) for every \(i=1,\ldots ,m\) . In particular, we employ the character sum method to attack this problem, which leads us in a natural manner to study novel incomplete character sum estimates and obtain results for fields of high order.
