The number of solutions of diagonal cubic equations over finite fields

Abstract

Let Fq be a finite field of q=pk elements. For any z∈Fq, let An(z) and Bn(z) denote the number of solutions of the equations x13+x23+⋯+xn3=z and x13+x23+⋯+xn3+zxn+13=0 respectively. Recently, using the generator of Fq⁎, Hong and Zhu gave the generating functions ∑n=1∞An(z)xn and ∑n=1∞Bn(z)xn. In this paper, we give the generating functions ∑n=1∞An(z)xn and ∑n=1∞Bn(z)xn immediately by the coefficient z. Moreover, we gave the formulas of the number of solutions of equation a1x13+a2x23+a3x33=0 and our formulas are immediately determined by the coefficients a1,a2 and a3. These extend and improve earlier results.