  Let $F(t),G(t)\in \mathbb{Q}(t)$ be rational functions such that $F(t),G(t)$
and the constant function $1$ are linearly independent over $\mathbb{Q}$, we
prove an asymptotic formula for the number of the three term rational function
progressions of the form $x,x+F(y),x+G(y)$ in subsets of $\mathbb{F}_p$. The
main new ingredient is an algebraic geometry version of PET induction that
bypasses Weyl's differencing. This answers a question of Bourgain and Chang.
