The paper is concerned with bounds for the moments of \[ W(a;q)=W_{f}(a;q)=\frac{1}{\sqrt{q}} \sum_{x\pmod q} e\left(\frac{af(x)}{q}\right), \] where \(f(x)\in {\mathbb Z}[x]\) is non-constant of degree \(d\), \(q\ge 1\) is squarefree and coprime to \(a\) (and \(W(a;q)=0\) in the remaining cases) and \(e(x)=e^{2\pi i x}\). By Weil’s bound, \(|W(a;q)|\le (d-1)^{\omega(q)}\). The paper is concerned with getting better bounds than the above on the average over \(q\) once \(a\) and \(f(x)\) are fixed. Theorem 1.1 asserts that if \(f\) is indecomposable (it cannot be written as a composition of two polynomials each of degree \(>1\)) then \[ \sum_{q\le x} |W(a;q)|^2 \ll x(\log\log x)^{(d-1)^2}, \] and there exists \(\gamma>0\) depending only on \(d\) such that \[ \sum_{q\le x} |W(a;q)|\ll \frac{x}{(\log x)^{\gamma}}. \] Theorem 1.3 shows that if \(d\ge 3\) then either \[ \lim_{p\to\infty} \frac{1}{p} \sum_{a\in {\mathbb F}_p^*} |W(a;p)|^4=2, \] or there exists \(\delta>0\) depending only on \(d\) and a subset of primes of density \(\ge \delta\) on which \[ \frac{1}{p}\sum_{a\in {\mathbb F}_p^*} |W(a;p)|^4\ge 3+O(p^{-1/2}). \] Theorem 1.11 proves non-correlation results for sums of \(W_{f_i}(a;q)\) for different polynomials \(f_1(x),\ldots,f_m(x)\). Under certain hypothesis on the polynomials (like that they are Sidon-Morse over \({\mathbb Q}\) and any two not linearly equivalent; that is, for \(i\ne j\), \(f_i(x)\) is not equal to \(af_j(cx+d)+b\) for some \(a,b,c,d\in {\overline{\mathbb Q}}\)) the authors obtain good upper bounds on \[ \sum_{q\le x} |W_{f_1}(a;q)\cdots W_{f_m}(a;q)|^s \] for \(s\in \{1,2,4\}\). The proofs use a theorem of Fried on the irreducibility of \((f(X)-f(Y))/(X-Y)\) as well as a theorem of Katz on the monodromy of the normalized sheaf of a Sidon-Morse polynomial \(f(x)\).
For the entire collection see [Zbl 1506.46001].