Let \(p\) be a prime, let \(f(x)=a_1x^{k_1} + \dots + a_rx^{k_r}\) (\(a_i, k_i \in {\mathbb Z}\)) be a Laurent polynomial, and let \(\chi\) be a multiplicative character mod \(p\). The authors obtain several new Mordell-type bounds for the mixed exponential sum \[ S(\chi, f) = \sum_{x=1}^{p-1} \chi(x) \exp(2\pi i f(x)/p), \] which are particularly effective when \(\gcd(k_i, p-1)\) is large for several values of \(i\). When \(p \nmid a_1 \cdots a_r\) and the \(k_i\) are distinct mod \(p-1\), bounds stemming from the methods of A. Weil [Proc. Natl. Acad. Sci. USA 34, 204–207 (1948; Zbl 0032.26102)] take the shape \(|S(\chi, f)| \leq D p^{1/2}\), where \(D=\max\{|k_i|, |k_i-k_j|\}\).
The most general versions of the authors’ results are rather lengthy to state, so we instead mention two special cases of interest. Write \(f_1(x)=ax^k + h(x^d)\) and \(f_2(x)=ax^k+bx^{-k}+h(x^d)\), where \(d\mid (p-1)\), \((k,p-1)=1\), \(p \nmid ab\), and where \(h\) is any Laurent polynomial. Then the authors prove in particular that \[ |S(\chi,f_1)| \leq p d^{-1/2} \quad \text{and} \quad |S(\chi, f_2)| \leq p d^{-1/2} + \sqrt{2} p^{3/4}. \] One strategy involves relating bounds for \(S\) to the number of solutions of associated systems of congruences and applying a result of T. D. Wooley [J.Aust.Math.Soc.88, No. 2, 261–275 (2010; Zbl 1231.11042)]. In another approach, the authors establish a reduction formula that successively exploits collections of exponents having a common factor before benignly inserting the Weil bound at the final stage.