将Igusa关于指数和与单值群的猜想与最小指数的半连续性相结合
Combining Igusa's conjectures on exponential sums and monodromy with semi-continuity of the minimal exponent
AI总结：
本研究结合Igusa的两个猜想与Mustaţă和Popa的半连续性结果，提出关于指数和界的新自然猜想，提供多类证据并证明其可推出Igusa强单值群猜想推论，关联圆法优弧估计。
摘要：
我们将Igusa的两个猜想与Mustaţă和Popa近期的半连续性结果相结合，提出了一个关于指数和界的全新自然猜想。这些界仅用次数和维度表述，形式看似简单却极具一般性。我们提供的证据部分改编自已有Igusa指数和猜想的相关结果，也包含新证据，例如对至多4个变量的所有多项式均成立的结论。我们进一步证明，这些界反过来能推出Igusa（强）单值群猜想的相关推论，且该界与圆法中用于局部-整体原理的优弧估计相关。
英文摘要：
We combine two of Igusa's conjectures with recent semi-continuity 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.
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