Weil bound for character sums

Question

I know that the well-known Weil bound for character sums is given by $$\left| \sum_{x \in \mathbb{F}_{q}} \chi(f(x)) \right| \leq (d-1)\sqrt{q}$$ where $\mathbb{F}_q$ is a finite field of size $q$, $\chi$ is a multiplicative character of order $m$, $f\in\mathbb{F}_q[x]$ is a polynomial of positive degree that is not an $m$th power of a polynomial, and $d$ is the number of distinct roots of $f$ in $\bar{\mathbb{F}}_q$. I also know that this bound can be generalized to rational polynomial $f=g/h$. Now I want to know if there exists any bound for $$\left| \sum_{x \in \mathbb{F}_{q}} \chi_1(x)\chi_2(f(x)) \right|$$ where $\chi_1$, $\chi_2$ are two non-principal multiplicative characters, and $f=g/h$ is a rational polynomial. Would anyone please give me some references?

Answer 1

If $\chi$ is a generator of $\widehat{\Bbb{F}_q^*}$, i.e. a multiplicative character of order $q-1$, then you can write $\chi_i=\chi^{d_i}$ for some exponents $1\le d_1,d_2<q-1$. It follows that $$ \chi_1(x)\chi_2(\frac{g(x)}{h(x)})=\chi(\frac{x^{d_1}g(x)^{d_2}}{h(x)^{d_2}}). $$ Looks like the bound you know about can be applied here. You just get one more zero if $x=0$ was not a zero of either $g(x)$ or $h(x)$ to begin with.