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 $$\sum _{n\le N}e(\alpha n^4){\ll }_{\text{ep},\alpha }{N}^{5\mathrm{/}6+\text{ep}}$$ for any $\text{ep}>0$ and any quadratic irrational $\alpha \in \mathbb{R}-\text{Q}$. Classically one would have had the exponent $7\mathrm{/}8+\text{ep}$ 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.