Abstract
Дан вариант метода Хуа для оценки неполных рациональных тригонометрических сумм. Эти оценки являются нетривиальными для сумм с длинами превосходящими корень квадратный от длины полной суммы.
Highlights
The purpose of this article is to give the new demonstration of the estimation of non-complete rational trigonometric sums
Let ξ be a solution of the congruence p−τ f ′(ξ) ≡ 0
We find pk−l−1 pk−l−1 e(−f (ξ)/pk)S(ξ) = ∑︁ e((f (ξ + px) − f (ξ))/pk) = ∑︁ e(g(x)/pk−u)
Summary
The purpose of this article is to give the new demonstration of the estimation of non-complete rational trigonometric sums. We develop the Hua’s method of estimations of complete rational sums ([2], p.101–109). Let n ≥ 2, p is a prime number, f (x) = anxn + · · · + a1x + a0 is a polynomial with integer coefficients, Let inequalities kr−1 ≥ 2(lr−1 + w + 1), kr < 2(lr + w + 1) be define the number r. Let r be the smallest number over all solutions Ξr), defining early, and satisfying inequalities kr−1 ≥ 2(lr−1 + w + 1), kr < 2(lr + w + 1). |S(pk; k − l, f )| ≤ (n − 1)pk−l−r
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