On the number of solutions of diagonal equations over a finite field
Let \(F\) be a finite field with \(q\) elements, \(q\) odd, and let \(c_i\in F\backslash \{0\}\), \(i= 1,\dots, n\), \(n> 1\), be fixed. For divisors \(d_i\) of \(q- 1\), let \(N(d_1,\dots, d_n)\) be the number of solutions \((x_1,\dots, x_n)\) of the equation \[ c_1 x^{d_1}_1+\cdots+ c_n x^{d_n}_n= 0. \] Let \(u_i:= \text{gcd}(d_i, d_1\cdot\ldots \cdot d_n/d_i)\). It is proved that \(N(d_1,\dots, d_n)= N(u_1,\dots, u_n)\), and furthermore if \(m_1,\dots, m_n\) are pairwise relatively prime odd integers, \(t\geq 0\), then \[ \begin{multlined} N(2m_1,\dots, 2m_{n- 1}, 2^t m_n)=\\ \begin{cases} q^{n- 1}+ (q- 1)^{(n- 2)/2} \eta((n- 1)^{n/2} c_1\cdot \cdots \cdot c_n)\quad & \text{if } 2\mid n,\;t> 0;\\ q^{n- 1}\quad & \text{if } 2\mid n,\;t= 0;\text{ or } 2\nmid n;\end{cases}\end{multlined} \] where \(\eta\) denotes the quadratic character of \(F\).
scientific article; zbMATH DE number 851860