Explicit values of multi-dimensional Kloosterman sums for prime powers, II

For any integer $m>1$ fix $\zeta_{m}={\rm exp}(2 \pi i/m)$, and let ${\bf Z}_{m}^{*}$ denote the group of reduced residues modulo $m$. Let $q=p^{\alpha}$, a power of a prime $p$. The hyper-Kloosterman sums of dimension $n>0$ are defined for $q$ by

$\displaylines{ R(d,q)= \sum_{x_{1}, ..., x_{n} \in Z_{q}^{*}} \ze... ...ots +x_{n} +d(x_{1} \cdots x_{n})^{-1}} \;\;\;\;\;\; (d \in {\bf Z}_{q}^{*}), }$

where $x^{-1}$ denotes the multiplicative inverse of $x$ modulo $q$.

Salie evaluated $R(d,q)$ in the classical setting $n=1$ for even $q$, and for odd $q=p^{\alpha}$ with $\alpha >1$. Later, Smith provided formulas that simplified the computation of $R(d,q)$ in these cases for $n>1$. Recently, Cochrane, Liu and Zheng computed upper bounds for $R(d,q)$ in the general case $n >0$, stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for $\alpha >1$, relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author's previous determination of these Kloosterman sums using character theory and $p$-adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.