# Bounds for general character sums over finite fields

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! $$\left| \sum_{x \in \mathbb{F}_q^*} \chi(f(x)) \psi(x) \right|$$

• Yes, look at the following Springer Lecture Note: W. M. Schmidt, Equations over finite fields: An elementary approach. Springer, Berlin (1976). (I used to have this book in my library until a few years ago....) – Dilip Sarwate Dec 19 '14 at 4:14
• Schmidt's book is very useful (our library copy is also mysteriously misplaced). Lidl & Niederreiter don't go into details here even though they explain the use of L-functions IMO a bit better. I benefitted from studying Michael Rosen's book in the sense that there is a more number theoretical account there. Unfortunately he doesn't go into details on the character sums. But you will recognize that those mysterious subgroups of the multiplicative group of $\Bbb{F}_q(x)$ are related to ray classes. – Jyrki Lahtonen Dec 19 '14 at 13:27
• Rosen does give Bombieri's version of the Schmidt-Stepanov method of proving the Riemann hypothesis for function fields. (If you are a coding theoretically minded, then a similar account is in Stichtenoth's book). Combining this with what you learn about L-functions from the other listed sources will go a long way. For example, the Shanbhag-Kumar-Helleseth extensions I refer to in my answer become "straightforward" exercises. – Jyrki Lahtonen Dec 19 '14 at 13:30

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 $$\left\vert\sum_{x\in\Bbb{F}_q}\chi(f_1(x)+f_2(\frac1x))\psi(x)\right\vert\le (\deg f_1+\deg f_2)\sqrt{q}.$$