We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.

中文翻译：

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IGUSA 的指数和猜想：非有理奇点的最佳估计

我们证明了满足特定幂条件的超曲面的对数规范阈值的上限，并使用它来证明 Igusa 猜想对指数和的几种推广，其中对数规范阈值在估计的指数中。我们通过将对数规范阈值与动力振荡指数的局部概念进行比较，表明这最佳地涵盖了非理性奇点猜想的所有情况。