## 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

### Keywords:

exponential sum; sparse polynomial; quadrinomial

### Citations:

Zbl 1082.11050; Zbl 1068.11053; Zbl 1448.11213
Full Text:

### References:

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