On a Conjecture of Chalk Original Research Article

Letfset membership, variantZ[x] with degreekand letpbe a prime. By a complete trigonometric sum we mean a sum of the formS(q, f)=∑qx=1 eq(f(x)), whereqis a positive integer andeq(α)=exp(2πif(x)/q). Professor Chalk made a conjecture on the upper bound ofS(q, f) whenqis a prime power. We prove Chalkʹs conjecture, in the affirmative, ifpis relatively small but greater-or-equal, slanted3. Whenpgreater-or-equal, slanted3 is relatively large, we give an alternative upper bound which is best possible. Forp=2, we also improve previous results.