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Bounds on exponential sums with quadrinomials. (English) Zbl 1450.11087

Let \(\chi\) be a multiplicative character and \(\psi\) be an additive character of the finite prime field \(F_p\). For a polynomial \(F(X)\in F_p[X]\) denote by \(S_\chi(F)\) the character sum \[S_\chi(F)=\sum_{x\in F_p^*}\chi(x)\psi(F(x)).\] The Weil bound \[|S_\chi(F)|\le \deg(F)p^{1/2}\] has been improved for sparse polynomials by various authors. For quadrinomials \(F(X)=aX^k+bX^\ell+cX^m+dX^n\) the author improves on these bounds, in particular on Theorem 1.1 of T. Cochrane et al. [Acta Arith. 116, No. 1, 35–41 (2005; Zbl 1082.11050)] and Theorem 1.1 of T. Cochrane and C. Pinner [Proc. Am. Math. Soc. 133, No. 2, 313–320 (2005; Zbl 1068.11053)]. He uses techniques from his joint work with I. D. Shkredov and I. E. Shparlinski [Can. J. Math. 70, No. 6, 1319–1338 (2018; Zbl 1448.11213)].

MSC:

11L07 Estimates on exponential sums
11T23 Exponential sums
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References:

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