Cluckers–Veys conjecture on exponential sums and local log canonical thresholds

Let $f\in\mathbf{Z}[x]$ be a non-constant polynomial in $n$ variables. For each $\alpha\in\mathbf{C}^n$, let $\sigma_\alpha(f)$ denote the log canonical threshold at $\alpha$ of the hypersurface $f(x)-f(\alpha)=0$, and define

$$S_f(p,m):=\frac{1}{p^{mn}}\sum_{x\in\{0,\ldots,p^m-1\}^n}\exp\left(2\pi\mathbf{i}\frac{f(x)}{p^m}\right).$$

Cluckers–Veys conjecture. There exists a constant $c\in\mathbf{R}_{>0}$ such that the bound

$$\lvert S_f(p,m)\rvert\leq c p^{-\left(\min_{\alpha\in\mathbf{C}^n}\sigma_\alpha(f)\right)m}m^{n-1}$$

holds for all primes $p$ and all integers $m\geq2$.

This proposed strengthening drops quasi-homogeneity when $m\geq2$ and replaces the threshold at the origin by the minimum of the relevant local thresholds. The text motivates this replacement by noting that the minimum can be strictly smaller than the threshold at the origin outside the quasi-homogeneous case; the conjecture is presented as a prediction of Cluckers and Veys.