UPPER BOUNDS ON n-DIMENSIONAL KLOOSTERMAN SUMS
نویسندگان
چکیده
Let pm be any prime power and Kn(a, pm) be the Kloosterman sum Kn(a, pm) = ∑pm x1=1 · · · ∑pm xn=1 epm (x1 + · · ·+ xn + ax1x2 . . . xn), where the xi are restricted to values not divisible by p. Let m, n be positive integers with m ≥ 2 and suppose that pγ‖(n + 1). We obtain the upper bound |Kn(a, pm)| ≤ (n + 1, p − 1)p 1 2 min(γ,m−2)pmn/2, for odd p. For p = 2 we obtain the same bound, with an extra factor of 2 inserted.
منابع مشابه
Explicit values of multi-dimensional Kloosterman sums for prime powers, II
For any integer m > 1 fix ζm = exp(2πi/m), and let Z ∗ m denote the group of reduced residues modulo m. Let q = pα, a power of a prime p. The hyper-Kloosterman sums of dimension n > 0 are defined for q by R(d, q) = ∑ x1,...,xn∈Z∗ q ζ x1+···+xn+d(x1···xn) q (d ∈ Zq), 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...
متن کاملA Note on Signs of Kloosterman Sums (une Note Sur Les Signes Des Sommes De Kloosterman)
We prove that the sign of Kloosterman sums Kl(1, 1;n) changes infinitely often as n runs through the square-free numbers with at most 15 prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distribu...
متن کاملRecursive formulas generating power moments of multi-dimensional Kloosterman sums and m-multiple power moments of Kloosterman sums
Abstract. In this paper, we construct two binary linear codes associated with multi-dimensional and m−multiple power Kloosterman sums (for any fixed m) over the finite field Fq. Here q is a power of two. The former codes are dual to a subcode of the binary hyper-Kloosterman code. Then we obtain two recursive formulas for the power moments of multi-dimensional Kloosterman sums and for the m-mult...
متن کاملSum-product Estimates in Finite Fields via Kloosterman Sums
We establish improved sum-product bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.
متن کاملCodes Associated with Special Linear Groups and Power Moments of Multi-dimensional Kloosterman Sums
In this paper, we construct the binary linear codes C(SL(n, q)) associated with finite special linear groups SL(n, q), with both n,q powers of two. Then, via Pless power moment identity and utilizing our previous result on the explicit expression of the Gauss sum for SL(n, q), we obtain a recursive formula for the power moments of multi-dimensional Kloosterman sums in terms of the frequencies o...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009