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In this paper, we will give two proofs of the Cluckers-Veys conjecture onexponential sums for the case of polynomials in$\mathbb{Z}[x_{1},\ldots,x_{n}]$ having log-canonical thresholds at most onehalf. In particular, these results imply Igusa's conjecture and Denef-Sperber'sconjecture under the same restriction on the log-canonical threshold.

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