AI 中文总结
该研究针对奇素数幂\textit{q}>204931的情况，结合特征和估计、改进素数筛法及精确有限计算，证明了Gow–McGuire关于本原二次多项式的猜想3，同时也证明了该范围内的猜想1和2。
AI 中文摘要
设\textit{q}为奇素数幂，\textit{μ}属于有限域\textit{F}\textsubscript{\textit{q}}的乘法群，\textit{α}属于有限域\textit{F}\textsubscript{\textit{q}\textsuperscript{2}}但不属于\textit{F}\textsubscript{\textit{q}}。我们研究族\textit{x}\textsuperscript{2}+\textit{μx}+\textit{λ}−\textit{α}中的本原多项式，其中\textit{λ}属于\textit{F}\textsubscript{\textit{q}}。根参数化将问题简化为求有理函数在\textit{q}+1个点上的本原值。结合特征和估计、改进的素数筛法及精确有限计算，我们证明了Gow和McGuire的猜想3对所有满足\textit{q}>204931的奇素数幂成立，且该范围内的猜想1和2也随之成立。
英文摘要
Let \(q\) be an odd prime power. For \(μ\in\mathbb F_q^\times\) and an additive coset \(\barα\in\mathbb F_{q^2}/\mathbb F_q\), consider the family \[ \{x^2+μx-α:α\in\barα\}, \] where each polynomial is regarded over its coefficient field \(\mathbb F_q(α)\). 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.
