EXPLICIT BOUNDS ON MONOMIAL AND BINOMIAL EXPONENTIAL SUMS

Let p be a prime and e(p)(.) = e(2 pi./p). First, we make explicit the monomial sum bounds of Heath-Brown and Konyagin: vertical bar Sigma(p-1)(x=1) e(p)(ax(d))vertical bar <= min{lambda d(5/8) p(5/8), lambda d(3/8) p(3/4)}, where lambda = 2/4 root 3 = 1.51967 .... Second, letting d = (k, l, p - 1), we obtain the explicit binomial sum bound vertical bar Sigma(p-1)(x=1) e(p)(ax(k) + bx(l))vertical bar <= (k - l, p - 1) + 2.292 d(13/46) p(89/92), for any non-constant binomial ax(k) + bx(l) on Z(p), by sharpening the estimate for the number of solutions of the system x(1)(k) + x(2)(k) = x(3)(k) + x(4)(k) and x(1)(l) + x(2)(l) = x(3)(l) + x(4)(l). Finally, we apply the latter estimate to establish the Goresky-Klapper conjecture on the decimation of l-sequences for p > 4.92 x 10(34).