On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field

In 1988 Garcia and Voloch proved the upper bound 4n4/3(p−1)2/3 for the number of solutions over a prime finite field Fp of the Fermat equation xn+yn=a, where and n⩾2 is a divisor of p−1 such that . This is better than Weil's bound in the stated range. By refining Garcia and Voloch's proof we show that the constant 4 in their bound can be replaced by 3⋅2−2/3.