On enumeration of irreducible polynomials and related objects over a finite field with respect to their trace and norm
Let \(\mathbb F_q\) be the finite field with \(q\) elements and let \(p=\text{char }\mathbb F_q\). For \(\gamma_1\in\mathbb F_q\), \(\gamma_2\in\mathbb F_q^*\) and \(m>0\), let \(I_m(\gamma_1,\gamma_2)\) denote the number of irreducible polynomials \(f(x)=x^m-a_{m-1}x^{m-1}+\cdots+(-1)^ma_0\in\mathbb F_q[x]\) with \(a_{m-1}=\gamma_1\) and \(a_0=\gamma_2\). The asymptotic formula for \(I_m(\gamma_1,\gamma_2)\), as \(m\to\infty\) and/or \(q\to\infty\), was a topic studied by many authors. The main term in the asymptotic formula for \(I_m(\gamma_1,\gamma_2)\) is known to be \(q^{m-1}/m(q-1)\). The objective of the present paper is to improve the previous bounds for the error term in the asymptotic formula for \(I_m(\gamma_1,\gamma_2)\). It is known that \[ I_m(\gamma_1,\gamma_2)=\frac 1m\sum_{d\mid m}\mu(m/d)\sum_{\alpha,\beta\in\mathbb F_q}N_d(\alpha,\beta)[\![(m/d)\alpha=\gamma_1,\,\beta^{m/d}=\gamma_2]\!], \] where \(\mu\) is the Möbius function, \[ N_d(\alpha,\beta)=|\{u\in\mathbb F_{q^d}:\text{trace}_{q^d/q}(\alpha)=\gamma_1,\; \text{norm}_{q^d/q}(\alpha)=\gamma_2\}|, \] and \[ [\![P]\!]=\begin{cases} 1&\text{if property \(P\) is true},\cr 0&\text{otherwise}. \end{cases} \] Moreover, \[ N_d(\gamma_1,\gamma_2)=\frac{q^{d-1}-1}{q-1}+(-1)^{d-1}\Bigl(M_d(\gamma_1,\gamma_2)-\frac{(q-1)^{d-1}-(-1)^{d-1}}q\Bigr), \] where \[ M_d(\gamma_1,\gamma_2)=\Bigl|\Bigl\{(\alpha_1,\dots,\alpha_d)\in{\mathbb F_q^*}^d:\sum_{j=1}^d\alpha_j=\gamma_1,\ \prod_{j=1}^d\alpha_j=\gamma_2\Bigr\}\Bigr|. \] A proof of the formula for \(N_d(\gamma_1,\gamma_2)\) using generating functions is included in the paper. The number \(M_d(\gamma_1,\gamma_2)\) can be expressed in terms of Gauss sums over \(\mathbb F_q\). The estimates for \(I_m(\gamma_1,\gamma_2)\) are obtained from the relation between \(I_m(\gamma_1,\gamma_2)\) and \(M_d(\gamma_1,\gamma_2)\) and the basic properties of Gauss sums. The main result (Theorem 2) gives two upper bounds for \(|I_m(0,\gamma_2)-A|\) and \(|I_m(\gamma_1,\gamma_2)-B|\) (\(\gamma_1\ne 0\)), respectively, where \(A\) and \(B\) contain more than just the main term in the asymptotic formula. In the corollaries, these bounds are simplified under additional assumptions. It is observed that these bounds improve the previous ones when \(m\) is large relative to \(q\).
scientific article; zbMATH DE number 7313134
scientific article