Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12]. In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of ℓ-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of $A-D-E$ type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

设f是某个数域上n 个变量的多项式，Z是仿射n空间的子方案。在Z处f的动力振荡指数的概念是由 Cluckers 在 [7] 和 Cluckers-Mustaţǎ-Nguyen 在 [12] 中提出的。在本文中，我们详细阐述了这个概念并提出了几个问题。第一个是基场扩展下的稳定性；这个问题是与非阿基米德局部域的密度的深刻理解，在其上井草的地方的zeta函数˚F有一个给定实部的极点。第二个是围绕 Igusa 的关于指数和的猜想，在动机振荡指数方面有界限。第三，我们想知道上述问题是否仅取决于奇点的解析同构类。通过使用 GAGA 定理、奇点解析和模型理论等各种技术，我们可以回答第三个问题直到基场扩展。接下来，通过使用特征为零的非阿基米德局部场和正特征的非阿基米德局部场之间的传递原理，我们可以将所有三个问题与对与射流多项式相关的 Artin-Schreier 层的ℓ -adic 上同调群的权重的猜想联系起来。这样，我们可以正面回答所有问题，如果f是具有非有理奇点的“Thom-Sebastiani 类型”多项式。因此，我们证明了三个变量中任意多项式的 Igusa 猜想和具有奇点的多项式$\mathrm{一个}-D-乙$类型。在附录中，我们肯定地回答了 [12] 中 Cluckers-Mustaţǎ-Nguyen 关于扭曲 Igusa 局部 zeta 函数的最大阶极点的最近问题。