Primitive Quadratic Polynomials in Additive Coset Families

Abstract

Let \(q\) be an odd prime power. For \(\mu\in\mathbb F_q^\times\) and an additive coset \(\bar\alpha\in\mathbb F_{q^2}/\mathbb F_q\), consider the family \[ \{x^2+\mu x-\alpha:\alpha\in\bar\alpha\}, \] where each polynomial is regarded over its coefficient field \(\mathbb F_q(\alpha)\). We prove that, for \[ q\notin{7,11,13,19,29,31,41,43}, \] every such family contains a primitive polynomial. For the zero coset, the result follows from Cohen's prescribed-trace theorem. For a nonzero coset, after a natural normalization the roots are parameterized by two smooth projective conics arising from the two \(q^2\)-Frobenius eigenspaces in \(\mathbb F_{q^4}\). Their affine \(\mathbb F_q\)-point counts, \(q-1\) and \(q+1\), correspond respectively to the reducible and irreducible members of the family. On the irreducible root conic, \(q^2\)-Frobenius induces a fixed-point-free involution on rational points. Passing to the quotient conic allows the relative norm of the root function to descend to a rational function. Tensor induction then yields order-sensitive character-sum bounds with constants \(6,8\) on the root conic and the sharper constants \(2,4\) after norm descent. Combining these estimates with a double-core prime sieve and an exact residue-cover verification completes the finite range. As consequences, Gow and McGuire's Conjecture~1 holds for every odd prime power \(q\ne13\), while their Conjectures~2 and~3 hold for every odd prime power \(q>43\); the threshold \(43\) is sharp.