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

@article{Mattarei2007OnAB,
title={On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field},
author={Sandro Mattarei},
journal={Finite Fields and Their Applications},
year={2007},
volume={13},
pages={773-777}
}

In 1988 Garćıa 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 a ∈ F∗p and n ≥ 2 is a divisor of p−1 such that (n− 1 2 ) 4 ≥ p− 1. This is better than Weil’s bound p + 1 + (n− 1)(n − 2)√p in the stated range. By refining Garćıa and Voloch’s proof we show that the constant 4 in their bound can be replaced by 3 · 2−2/3. Let Fq be the finite field of q elements and let p be its characteristic… CONTINUE READING

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