## 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:

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