Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772646 | Journal of Number Theory | 2017 | 34 Pages |
Abstract
We investigate the p-adic valuation of Weil sums of the form WF,d(a)=âxâFÏ(xdâax), where F is a finite field of characteristic p, Ï is the canonical additive character of F, the exponent d is relatively prime to |FÃ|, and a is an element of F. Such sums often arise in arithmetical calculations and also have applications in information theory. For each F and d one would like to know VF,d, the minimum p-adic valuation of WF,d(a) as a runs through the elements of F. We exclude exponents d that are congruent to a power of p modulo |FÃ| (degenerate d), which yield trivial Weil sums. We prove that VF,dâ¤(2/3)[F:Fp] for any F and any nondegenerate d, and prove that this bound is actually reached in infinitely many fields F. We also prove some stronger bounds that apply when [F:Fp] is a power of 2 or when d is not congruent to 1 modulo pâ1, and show that each of these bounds is reached for infinitely many F.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Daniel J. Katz, Philippe Langevin, Sangman Lee, Yakov Sapozhnikov,