Additive character sums of polynomial quotients (Q2869127)

Let \(p\) be a prime number, \(f(x) \in\mathbb Z[x]\) a polynomial with leading coefficient not divisible by \(p\), and \(R\) a complete residue system modulo \(p\). For an integer \(u\) the authors define the polynomial quotient \(q_{f,p,R}(u)\) by NEWLINE\[NEWLINE q_{f,p,R}(u)= \frac{f(u)-f_{p,r}(u)}{p} \pmod p, \quad 0 < q_{f,p,R}(u) < p, NEWLINE\]NEWLINE where \(f_{p,R}(u) \equiv f(u)\pmod p\), \(f_{p,R}(u) \in R\), and then find an upper bound for the additive character sums NEWLINE\[NEWLINE \sum_{u=M+1}^{M+N} \psi \left( \sum_{j=0}^{s-1}a_{j}q_{f,p,R}(u+j)\right), \quad 1 \leq s \leq \deg(f), NEWLINE\]NEWLINE which is non-trivial for any \(N \geq \deg(f)p \log p\). This extends a series of earlier results concerning the well-studied Fermat quotient NEWLINE\[NEWLINE q_{p}(u)= \frac{u^{p-1}-u^{p(p-1)}}{p} \pmod p. NEWLINE\]NEWLINE In addition, for \(s=1\) and NEWLINE\[NEWLINE q_{p}(u) \equiv \frac{u^{w}-u^{pw}}{p} \pmod pNEWLINE\]NEWLINE with a large \(\gcd(w,p-1)\) the authors obtain much stronger bounds by a reduction to the well-known Burgess bound.NEWLINENEWLINEFor the entire collection see [Zbl 1253.00023].