On the Maximum Modulus of Complete Trigonometric Sums

Let Q(p) be a set consisting of all polynomials of degree k with integral coefficientsf(x)=ax~k+…+ax,where pa.For given k and p any polynomial f(x)∈Q(p) satisfying|S(p,f)|=sup|S(p,f)|∈Q(p)is called a maximum modular polynomial in Q(p),whereS(p,f)=e~(2πi(x)/p).Moreover,we definec(k,p)=|S(p,f(x))|.The main results are the following theorems.Theorem 1.For k=p-1 and p≥3 we havec(k,p)=(p~2-4(p-1)sin~2π/p).Besides,we may take f(x)=(x-r).Theorem 2.For k=p-s,2≤s≤(p+1)/2 and p≥5,we havec(k,p)≤p-4(s-1)sin~2π/p.In Theorems 3 and 4,an interesting connextion between the present question and the famous problemof Prouhet and Tarry is given,some conditions under which the sign of equality in Theorem 2 holds aregiven and a method used to construct a maximum modular polynomial in Q(p) is also given.