On the Carlitz problem on the number of solutions to some special equations over finite fields (Q449687)

Let \(\mathbb F_q\) be a finite field with \(q=p^s\) elements and let \(N_q\) be the number of solutions in \(\mathbb F_q\) of \(\sum_{i=1}^n a_ix_i=bx_1\cdots x_n\). \textit{L. Carlitz} [Monatsh. Math. 58, 5--12 (1954; Zbl 0055.26803)] gave formulas for \(N_q\) when \(n=3\) and when \(n=4\), \(q\equiv 3\pmod 4\). In previous papers, the author found explicit formulas for \(N_q\) when \(d=(n-2, (q-1)/2)\) is \(1,2\) or \(4\). Here formulas are found when \(d=2^t\), \(t\geq 3\) and \(p\equiv 3,5\pmod 8\) or \(p\equiv 9\pmod{16}\). The results are obtained via Gauss sums and are written in terms of the quadratic character on \(\mathbb F_q\) and certain, uniquely determined, solutions to quadratic equations of the form \(q^k=x^2+2y^2\) or \(x^2+y^2\).