## 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

### Citations:

Zbl 0621.10024; Zbl 0731.11047
Full Text:

### References:

 [1] Loxton, Canad. Math. Bull. 28 pp 440– (1985) · Zbl 0575.10033 [2] Hua, Enzyklopädie der Math. pp 41– (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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