Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}\_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!



Minimal expression of 0 as a sum of kth powers in a finite field
Write $q=p^n$. You can replace $k$ by $gcd(k,q-1)$ and assume $k | (q-1)$. Now, trivially, $l=2$ if and only if $(-1)^{(q-1)/k} = 1$. If $l >2$, then $l=3$ if the Fermat curve has a point which, from the Weil bound, happens if $k < q^{1/4}$, approximately. For larger $k$ you may want to use techniques from additive combinatorics. Someone else may chip in on that.


