What is known about approximating a bivariate function by a sum of univariate functions e.g. approximating $a(x)+b(y)=f(xy)$ where $x,y \in \mathbb{F}_p$?
I have in mind probably simplest non-trivial scenario $a(x)+b(y)=-xy$. For example, one can check that in the case $\mathbb{F}_3$ the best approximation is achieved by something like $a(x)=\delta_{x,2},\ b(y)=\delta_{y,2}$. This is correct on 6/9 values $(x,y)\in\mathbb{F}_3^2$.
Does this problem reduce to something famous that would find me better search results i.e., is there better terminology to describe this?
Ideally, I would also like to do better than brute force in searching for optimal solutions.