On a Conjecture of Chalk

AbstractLetf∈Z[x] with degreekand letpbe a prime. By a complete trigonometric sum we mean a sum of the formS(q, f)=∑qx=1eq(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 Chalk's conjecture, in the affirmative, ifpis relatively small but ⩾3. Whenp⩾3 is relatively large, we give an alternative upper bound which is best possible. Forp=2, we also improve previous results