A system of simultaneous congruences arising from trinomial exponential sums
نویسندگان
چکیده
For a prime p and positive integers l < k < h < p with d = (h, k, l, p−1), we show that M , the number of simultaneous solutions x, y, z, w in Zp to xh + yh = zh + wh, xk + yk = zk + wk, xl + yl = zl + wl, satisfies M ≤ 3d(p− 1) + 25hkl(p− 1). When hkl = o(pd2) we obtain a precise asymptotic count on M . This leads to the new twisted exponential sum bound ∣∣∣∣∣ p−1 ∑ x=1 χ(x)e ∣∣∣∣∣ ≤ 3 4 d 1 2 p 8 +5(hkl) 1 4 p 5 8 , for trinomials f = axh + bxk + cxl, and to results on the average size of such sums.
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