A congruence equation in \(\mathrm{GF}[p^n,x]\) and some related arithmetical identities (Q2523590)

Let \(R\) be a primary (monic) polynomial of degree \(r\) in \(\Omega = \mathrm{GF}[p^n, x]\). Let \(r_1,r_2,s_1,s_2\) be any non-negative integers and \(A_i, B_j\) for \(0\leq i\leq s_1\), \(0\leq j\leq s_2\) be elements of \(\Omega\) such that \((A_i, R) = (B_j, R) = 1\) for all \(i, j\). Let \(N = N_{r_1,r_2}^{(s_1,s_2)}) (A, R)\) denote the number of solutions \(X_{ij}, Y_{kp}\) for \(1\leq i\leq r_1+1\), \(0\leq j\leq s_1\), \(1\leq k\leq r_2+1\), \(0\leq p\leq s_2\) such that \((Y_{kp}, R) = 1\) for all \(k, p\) and \[ A\equiv \sum_{j=0}^{s_1} A_j \prod_{i=1}^{r_1+1} X_{ij} + \sum_{p=0}^{s_1} B_p \prod_{k=1}^{r_1+1} Y_{kp} \pmod R. \] If \(F\in\Omega\) and \(F\equiv \sum_{i=1}^r \alpha_i X^{r-i} \pmod R\) with \alpha_i\in \mathrm{GF}(p^n)\(, let \)\varepsilon(F) = \varepsilon(F, R) = \exp (2\pi i\alpha_1/p)\(, where \)\alpha_1\in\sum_{j=1}^n \alpha_j \vartheta^{n-1}\( and \)\vartheta\( generates \)\mathrm{GF}(p^n)\() over \)\mathrm{GF}(p)\(. The author obtains a recursion formula for \)N\( which involves {\mathit L. Carlitz}\,'s \)\eta\(-sum [Duke Math. J. 14, 1105--1120 (1947; Zbl 0032.00204)] \)\eta(A, R) = \sum_{(Z,R)} \varepsilon(ZA)\( and the Euler \)\varphi\(-function for \)\Omega\(. The proof uses the unique representation of arithmetic functions \)\pmod R\( on \)\Omega\(, in terms of the \)\eta\(-sum, proved first by Carlitz [Duke Math. J. 14, 1121--1137 (1947; Zbl 0032.00301)] and the formula proved by {\mathit Eckford Cohen} [Proc. Am. Math. Soc. 3, 352--358 (1952; Zbl 0049.15205)] giving the representation of the Cauchy product of two such functions in terms of the representations of the factors. Several identities involving \)N\( are also obtained by use of Cauchy products. For example, \)\(\sum_{A\pmod R}N_{r_1,r_2}^{(s_1,s_2)}) (A, R) = (p^{nr})^{(r_1+1) (s_1+1)} [\varphi(R)]^{(r_2+1)(s_2+1)}.\) \[ \]