  In this paper we prove an explicit version of a function field analogue of a
classical result of Odoni 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 of Bary-Soroker, Smilanski, and Wolf which deals with finding
asymptotics for a function field version on sums of two squares, improved upon
by Gorodetsky , 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 work of Church, Farb and Ellenberg on representation stability
and asymptotic for point counts on varieties. 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.
