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A further refinement of Mordell’s bound on exponential sums. (English) Zbl 1082.11050
Let \(p\) be an odd prime, and consider the Laurent polynomial \(f(x)=a_1 x^{k_1} + \ldots + a_r x^{k_r},\) where \(p\nmid a_i\) and the \(k_i\) are non-zero mod \(p-1\). Let \(\chi\) be a multiplicative character mod \(p\). The authors consider bounds for the mixed exponential sum \(S(\chi;f)= \sum_{x=1}^{p-1} \chi(x) e_p(f(x)).\) The classical Weil bound is \(| S(\chi;f)| \leq dp^{1/2}\), where \(d\) is the degree of \(f\) if \(f\) is a polynomial, and the degree of the numerator when \(f\) has both positive and negative exponents. The Weil bound is trivial when \(d\geq \sqrt p\). L. J. Mordell [Q. J. Oxf., Ser. 3, 161–167 (1932; Zbl 0005.24603)] gave a different bound that depended on the product of the exponents \(k_i\). In an earlier paper [Proc. Am. Math. Soc. 133, 313–320 (2005; Zbl 1068.11053)] the first and third authors obtained an improvement of Mordell’s bound; they showed that \[ | S(\chi;f)| \leq 4^{1/r} (\ell_1 \ldots \ell_r)^{1/r^2} p^{1-1/2r} \tag{*,} \] where \(\ell_i=k_i\) if \(k_i\) is positive and \(\ell_1=r| k_i| \) if \(k_i\) is negative. This is nontrivial if \((\ell_1 \ldots \ell_r) \leq 4^{-r} p^{r/2}\). In this article, the authors obtain a further improvement; in particular, they show some of the large values of \(\ell_1\) may be omitted from the product in the bound. Their new bound is \[ | S(\chi,f)| \leq 4^{1/m} (\ell_1 \ldots \ell_m)^{1/m^2} p^{1-(m-r/2)/m^2}, \] where \(m\) is any integer with \(r/2 < m \leq r\) and \(\ell_i=k_i\) if \(k_i\) is positive, \(\ell_i=m| k_i| \) if \(k_i\) is negative. For trinomials, i.e. \(f(x)= a_1 x^{k_1}+ a_2 x^{k_2} + a_3 x^{k_3}\), \(0 < k_1 < k_2 < k_3\), their bound is \(| S(\chi,f)| \leq 2(k_1 k_2)^{1/4} p^{7/8}\), which is independent of \(k_3\). All of the above bounds depend on upper bounds for the number of solutions of the simulataneous equations \[ x_1^{k_i} + \ldots + {x_m}^{k_i} = y_1^{k_i} + \ldots + {y_m}^{k_i} \pmod p \] for \(i=1,\ldots, r\). The crucial observation is that in some cases it is more efficient to use only the first \(m\) equations and discard the remaining \(r-m\) equations.

MSC:
11L07 Estimates on exponential sums
11L03 Trigonometric and exponential sums (general theory)
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