An upper bound for the extended Kloosterman sums over Galois rings (Q1273725)

For a fixed prime \(p\) and integers \(e\geq 2\), \(m\geq 1\), write \(R_{e,m}\) for the Galois ring of characteristic \(p^e\) and containing \(p^{em}\) elements. Let \(\psi_{e,m}:R_{e,m}\to{\mathcal C}\) be an additive character of \(R_{e,m}\) and let \(f(x)\) be a nondegenerate polynomial over \(R_{e,m}\) with weighted degree \(D_{e,f}\). In 1995, Kumar and others obtained an expression for sums of the form \(\Sigma\psi_{a,m} (f(x))\), where the summation runs over the set of so-called Teichmüller representatives \({\mathcal T}_{e,m}\), leading to a certain bound for the absolute value of the sum. In the present paper, the authors obtain an upper bound for the so-called extended Kloosterman sums over Galois rings, i.e. exponential sums of the form \(\Sigma \psi_{e, m}(f_1(n)+ f_2(n^{-1}))\), \(f_1,f_2\in R_{e,m} [x]\), the summation running over \({\mathcal T}^*_{e,m}\), the non-zero elements of \({\mathcal T}_{e,m}\) referred to above. Some applications to sequence designs are also given.