The Chevalley-Warning and the Ax-Katz theorems, which provide the p-adic valuations for the number of solutions of the system of polynomials in finite fields, are stated in terms of the degrees of the associated polynomials. Recently, Clark et al. reformulated the theorems in terms of the cardinalities of fibers and images of the associated polynomial functions, and raised a question in this setting. In this paper, we give a negative answer to that question and improve their results in general case using p-adic lifting via the Teichmüller character.

中文翻译：

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与多项式函数的纤维和图像相关的 p-Adic 估值

Chevalley-Warning 和 Ax-Katz 定理为有限域中多项式系统的解数提供p- adic 估值，以相关多项式的次数表示。最近，克拉克等人。根据纤维的基数和相关多项式函数的图像重新表述定理，并在这种情况下提出了一个问题。在本文中，我们对这个问题给出了否定的答案，并通过 Teichmüller 特征使用p- adic 提升在一般情况下改进了他们的结果。