Explicit evaluation of certain exponential sums of binary quadratic functions

Let 0<alpha(1) <... < alpha(k) be integers and f(x) = Sigma(k)(i=1) a(i)x(2 alpha i+1) +bx is an element of F-2m[x], a(k) not equal 0. Define S(f,n)= Sigma(x is an element of F2n) e(f(x)) where m vertical bar n and e(x) =(-1)(TrF)2(n/F2(x)). We establish a relation among S(f,n) for all n with the same 2-adic order. When nu(2)(alpha(1)) =...= nu(2)(alpha(k)), where nu(2) is the 2-adic order function, we are able to compute S(f, n) explicitly for all n with a given f. Moreover, we are able to compute S(ax(2 alpha) + 1 + cx, n) explicitly for all alpha > 0, a is an element of F-2m, m vertical bar n and c is an element of F-2n. (c) 2006 Elsevier Inc. All rights reserved.