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 $$\sum_{n\le N} e(\alpha n^4)\ll_{\ep,\alpha}N^{5/6+\ep}$$ for any $\ep>0$ and any quadratic irrational $\alpha\in\R-\Q$. Classically one would have had the exponent $7/8+\ep$ for such $\alpha$. In contrast to the author's earlier work \cite{cubweyl} 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.