On diagonal equations over finite fields (Q5919128)

Let \(q=p^n\), where \(p\) is an odd prime and \(n\) is a positive even integer. Let \(t_1,\dots,t_s\) be positive integers such that \(2t_i\mid n\) for all \(1\le i\le s\). Consider the diagonal equation \[ a_1x_1^{d_1}+\cdots+a_sx_s^{d_s}=b, \] where \(b\in\mathbb F_q\), \(a_i\in\mathbb F_q^*\), and \(2\le d_i\mid p^{2t_i}-1\). Let \(\vec a=(a_1,\dots,a_s)\), \(\vec d=(d_1,\dots,d_s)\), \(\vec t=(2t_1,\dots,2t_s)\), and let \(N_s(\vec a, \vec d,\vec t,q,b)\) denote the number of solutions \((x_1,\dots,x_s)\) of the above equation with \(x_i\in\mathbb F_{p^{2t_i}}\). The main objective of the paper is to develop explicit formulas for this number under certain conditions on the exponents \(d_1,\dots,d_s\). For positive integers \(d\) and \(r\), \(d\) is said to be {\em \((p,r)\)-admissible} if \(r\) is the smallest positive integer such that \(d\mid p^r+1\). When \(d=2\), \(d\) is \((p,1)\)-admissible. When \(d>2\), \(d\) is \((p,r)\)-admissible if and only if the multiplicative order of \(p\) in \(\mathbb Z/d\mathbb Z\) is \(2r\). Under the assumption that each \(d_i\) is \((p,r_i)\)-admissible, explicit formulas for \(N_s(\vec a, \vec d,\vec t,q,0)\) and for \(N_s(\vec a, \vec d,\vec t,q,b)\) with \(b\in\mathbb F_q^*\) and \(d_1=\cdots=d_s=d\) are derived (Theorem~2.4 and Corollary~2.6). An estimate for \(N_s(\vec a, \vec d,\vec t,q,0)\) is obtained (Corollary~2.7). The case with \(\vec t=(n,\dots,n)\), where \(N_s(\vec a, \vec d,\vec t,q,b)\) is denoted by \(N_s(\vec a,\vec d,q,b)\), is covered in Theorem~2.9, Corollary~2.10 and Theorem~2.11. These formulas generalize some previous results by other authors on the number of solutions of diagonal equations over finite fields. The computations in the paper largely rely on two facts: 1. \(\text{Tr}_{q/p}(ax^{p^r+1})\), where \(a\in\mathbb F_q^*\), is a quadratic form and the number of solutions of the equation \(\text{Tr}_{q/p}(ax^{p^r+1})=\lambda\) is known. 2. The Gauss quadratic sum and certain Jacobi sums are known. When \(d_1=\cdots=d_s=d>2\), \((s,b)\ne(2,0)\), \((s,d,b)\ne(4,3,0)\) and \((s,d)\ne(3,3)\) if \(b\ne 0\), it is shown that if \(N_s(\vec a,\vec d,q,b)\) attains the upper or lower bounds by Weil, then \(d\) is \((p,r)\)-admissible for some positive integer \(r\). Moreover, under these assumptions, necessary and sufficient conditions in terms of \(\vec a\), \(d\), \(n\), \(r\), \(b\) are found for \(N_s(\vec a,\vec d,q,b)\) to be maximal or minimal (Theorem~2.14).