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On Hua’s lemma. (English) Zbl 0833.11040

Let \(f(X)\) denote a polynomial of degree \(k\) with integer coefficients and let \[ S(p^\ell, f):= \sum_{0\leq x< p^\ell} \exp (2\pi ip^{-\ell} f(x)), \] where \(p\) is a prime and \(\ell\geq 2\). For a polynomial \(g\in \mathbb{Z} [X]\), let \(\nu_p (g)\) denote the \(p\)-content of \(g\). Suppose \(\nu_p (f(X)- f(0))= 0\). Hua proved that \[ |S(p^\ell, f)|\leq k^3 p^{\ell (1- 1/k)}. \] This estimate was subsequently sharpened by J. H. H. Chalk [Mathematika 34, 115-123 (1987; Zbl 0621.10024)] and P. Ding [Acta Arith. 59, 149-155 (1991; Zbl 0731.11047)] respectively. In this paper, the author obtains a further improvement on this. The argument is based on that of Chalk, using induction on \(\ell\), but uses a more precise form of a main lemma of Hua.
Let \(\nu_p (f' (X)) =t\) and let \(r= r(f)\) be the number of distinct roots of the congruence \(p^{-t} f' (X)\equiv 0\pmod p\). Let \(m_1, m_2, \dots, m_r\) be the multiplicities of these roots and let \(m= m_1+ m_2+ \cdots+ m_r\), \(M= \max (m_1, m_2, \dots, m_r)\).
Theorem 1. Let \(p\leq k\) and \(\theta (p)=1\) or 2 according as \(p\geq 3\) or \(p=2\).
(i) If \(r(f)>0\), then \[ |S(p^\ell, f)|\leq mp^{(t+ \theta)/ (M+1)} p^{\ell (1- 1/(M+ 1))}; \] (ii) if \(r(f)= 0\), then \(S(p^\ell, f) =0\) for \(\ell> t+ \theta\) and \(|S(p^\ell, f)|\leq p^{t+\theta}\) otherwise.
Theorem 2. Let \(p> k\geq 2\). Suppose \(r(f)>0\). Then \[ |S(p^\ell, f)|\leq mp^{\ell(1- 1/(M+ 1))}. \]

MSC:

11L07 Estimates on exponential sums
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References:

[1] Loxton, Canad. Math. Bull. 28 pp 440– (1985) · Zbl 0575.10033
[2] Hua, Enzyklopädie der Math. pp 41Р(1959)
[3] Chalk, Mathematika 34 pp 115– (1987)
[4] Ding, Acta Arith. LIX pp 149– (1991)
[5] Hua, Additive theory of prime numbers pp 2– (1965)
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