We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some new evidence like for all polynomials in up to $4$ variables. We show that, in turn, these bounds imply consequences for Igusa’s (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles.

我们将 Igusa 的两个猜想与 Mustaţă 和 Popa 最近的半连续性结果结合起来，形成一个关于指数和界限的新的、自然的猜想。这些界限仅在度数和维度方面有一个看似简单且通用的公式。我们提供的证据部分包括对 Igusa 指数和猜想的已知结果的改编，但也提供了一些新证据，例如最多 $4$ 变量的所有多项式。我们证明，反过来，这些界限意味着伊草（强）单一性猜想的后果。这些界限与局部-全局原理的圆法中出现的主要弧的估计有关。