Bounds for general character sums over finite fields

Question

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $\chi$ be the canonical additive character of $\mathbb{F}_q$, let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$, and let $f \in \mathbb{F}_q[x]$ be a polynomial. Is an upper bound known for the following general character sum, say in the style of Weil's bound? Thanks! $$ \left| \sum_{x \in \mathbb{F}_q^*} \chi(f(x)) \psi(x) \right| $$

Answer 1

Yes, there is such a bound. Below I list a few variants. To avoid trivial cases we need to assume that $f(x)$ is not of the form $h(x)^p-h(x)+r$ for any polynomial $h(x)$ and any constant $r$. I also include sums where the multiplicative also has a polynomial argument. Then non-triviality condition takes the form that if $\psi$ is of order $d$ the polynomial $g(x)$ should not be of the form $r h(x)^d$, again for any constant $r$ and any polynomial $h(x)$. The definition of a non-trivial multiplicative character is often extended by declaring that $\psi(0)=0$.

A general bound for hybrid sums with polynomial arguments is $$ \left\vert\sum_{x\in\Bbb{F}_q}\chi(f(x))\psi(g(x))\right\vert\le (\deg f+\deg g-1)\sqrt{q}. $$
The sums you specifically asked about are a special case of this with $g(x)=x$ of degree $1$.

Occasionally useful generalizations involve Laurent polynomials and read $$ \left\vert\sum_{x\in\Bbb{F}_q}\chi(f_1(x)+f_2(\frac1x))\right\vert\le (\deg f_1+\deg f_2)\sqrt{q} $$ and $$ \left\vert\sum_{x\in\Bbb{F}_q}\chi(f_1(x)+f_2(\frac1x))\psi(x)\right\vert\le (\deg f_1+\deg f_2)\sqrt{q}. $$

Other generalizations have been developed for analogous sums over Galois rings. Ask if you need to know more. Shanbhag, Kumar and Helleseth extended the calculations from Schmidt's book (see Dilip Sarwate's comment) to cover those cases. Wei-Ching (Winnie) Li gave more class field-theoretical proofs and some generalizations (such as cases where the summation is only over a subfield).