Improved Weyl bounds on short intervals

For an integer $d\ge 3$, put ${\mathrm{\Delta }}_d=\min 2^{d-1},d(d-1)$. Let $a\mathrm{/}q$ be reduced, let $P(X)=\frac{a}{q}{X}^d+{\alpha }_{d-1}{X}^{d-1}+\cdots +{\alpha }_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\mid {\sum }_{n\in \mathcal{I}}e(P(n))\mid {\ll }_{d,\epsilon }{q}^{1\mathrm{/}d}{H}^{\epsilon }+H^{1-1\mathrm{/}{\mathrm{\Delta }}_d+\epsilon }$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in \mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1\mathrm{/}d}<H<p^{1\mathrm{/}(d-1)}$, writing $H^d\mathrm{/}p=H^u$, one has $\mid \sum n\in \mathcal{I}{e}_p(P(n))\mid {\ll }_{d,\epsilon }{H}^{1-\min u\mathrm{/}d,1\mathrm{/}{\mathrm{\Delta }}_d+\epsilon }$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.