Bounds for the Quartic Weyl Sum

We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that∑n≤Ne(αn4)≪ε,αN5/6+ε for any ε>0 and any quadratic irrational α∈R−Q. Classically one would have had the exponent 7/8+ε for such α. In contrast to the author's earlier work [2] on cubic Weyl sums (which was conditional on the abc-conjecture), we show that the van der Corput AB-steps are sufficient for the quartic case, rather than the BAAB-process needed for the cubic sum.