In this paper, we prove an explicit version of a function field analogue of a classical result of Odoni (Mathematika 22(1):71–80, 1975) about norms in number fields, in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result Bary-Soroker et al. (Finite Fields Appl 39:195–215, 2016) which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky (Mathematika 63(2):622–665, 2017), and reproved by the author in his Ph.D. thesis using the method of this paper. The main tool is a twisted Grothendieck–Lefschetz trace formula, inspired by the paper (Church et al. in Contemp Math 620:1–54, 2014). Using a combinatorial description of the cohomology, we obtain a precise quantitative result which works in the \(q^n\rightarrow \infty \) regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.

在本文中，我们证明了在循环Galois扩展的情况下有关数域范数的Odoni（Mathematika 22（1）：71–80，1975）的经典结果的函数域类似物的显式版本。在二次扩展的特定情况下，我们恢复了Bary-Soroker等人的结果。（Finite Fields Appl 39：195–215，2016），该函数处理两个平方和求函数域版本的渐近性，由Gorodetsky改进（Mathematika 63（2）：622–665，2017），并由作者在他的博士学位 论文采用本文的方法。主要工具是受本文启发的扭曲的Grothendieck-Lefschetz跟踪公式（Church等人，Contemp Math 620：1-54，2014）。使用对同调的组合描述，我们获得了精确的定量结果，该结果在\（q ^ n \ rightarrow \ infty \）体制，以及一种新型的同态稳定现象，它是由表示的某些内积的计算产生的。