ON THE EQUATION $(x_1^{m_1}+\cdots+x_n^{m_n})^k=ax_1\cdots x_n$ OVER A FINITE FIELD
Ioulia N. Baoulina · International Journal of Number Theory · 2006
Let Nq be the number of solutions to the equation [Formula: see text] over the finite field 𝔽q = 𝔽ps. Carlitz found formulas for Nq when m1 = ⋯ = mn = 1, k = 2, n = 3 or 4, p > 2; and when m1 = ⋯ = mn = 2, k = 1, n = 3 or 4, p > 2. In earlier papers, we obtained some generalizations of Carlitz's results. In this paper, we find formulas for Nq when -1 is a power of p modulo dD, where D = lcm [d1,…,dn], [Formula: see text], M = lcm [m1,…,mn], dj = gcd (mj,q - 1), 1 ≤ j ≤ n.