On the maximum modulus of complete trigonometric sums |
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Authors: | Zhang Mingyao Hong Yi |
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Affiliation: | 1. Institute of Mathematics, Academia Sinica, China
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Abstract: | LetQ k (p) be a set consisting of all polynomials of degreek with integral coefficientsf(x)=a k x k +...+a 1 x, wherep×a k . For givenk andp any polynomialf k,p (x)εQ k (p) satisfying ‖S(p, f k,p )‖=sup ‖S(p, f)‖fεQ(p) is called a maximum modular polynomial inQ k (p), where $$S(p,f) = sumlimits_{x = 0}^{p - 1} {e^{2pi if(x)/p} } $$ Moreover, we definec(k, p)=‖S(p, f k.p (x))‖. The main results are the following theorems. - For k=p?1 and p≥3 we have $$c(k,p) = sqrt {p^2 - 4(p - 1)sin ^2 frac{pi }{p}} $$ Besides, we may take (f_{k,p} (x) = prodlimits_{r = 0}^{p - 2} {(x - r)} )
- For k=p?s, 2≤s≤(p+1)/2 and p≥5, we have $$c(k,p) leqslant p - 4(s - 1)sin ^2 frac{pi }{p}$$ .
In Theorems 3 and 4, an interesting connextion between the present question and the famous problem of Prouhet and Tarry is given, some conditions under which the sign of equality in Theorem 2 holds are given and a method used to construct a maximum modular polynomial inQ k (p) is also given. |
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