Proof of Cluckers-Veys's conjecture on exponential sums for polynomials with log-canonical threshold at most a half

In this paper, we will give two proofs of the Cluckers-Veys conjecture on exponential sums for the case of polynomials in $\mathbb{Z}[x_1,\dots ,x_n]$ having log-canonical thresholds at most one half. In particular, these results imply Igusa’s conjecture and Denef-Sperber’s conjecture under the same restriction on the log-canonical threshold.