## 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)

exponential sums

### Citations:

Zbl 0005.24603; Zbl 1068.11053
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