Consider the absolute value of the following exponential sum:
$\left|\sum_{x \in \mathbb{F}_p} \sum_{y \in \mathbb{F}_p}e^{\frac{2\pi i}{p}(ux+vy-wxy)}\right|$
for given $u,v,w\in\mathbb{F}_p$ with $w \neq 0$.
Through computer experimentation I have observed that this is always equal to $p$, but I am unable to prove this for the case where both $u \neq 0$ and $v \neq 0$.
Why is this true? And if it is not, is there a counterexample?
Expand $(1+ax)(b+vy)$ and identify with $b+ux+vy-wxy$, assuming that $u\ne 0,v\ne 0$.
So $a\ne 0,v\ne 0$ and it is immediate that $$\sum_{x \in \mathbb{F}_p} \sum_{y \in \mathbb{F}_p}e^{\frac{2\pi i}{p}(1+ax)(b+vy)} =p$$