Decomposition of bivariate polynomials over finite fields as a sum of univariate products

Question

Let $p$ be a prime. Given a bivariate polynomial $f(X,Y)\in \mathbb{F}_p[X,Y]$ with degrees $d_1,d_2$ in $X,Y$ respectively, what is the lowest known upper bound on the smallest integer $k$ such that $f(X,Y)$ can be expressed as a sum of products $$ f(X,Y) = \sum\limits_{i=1}^{k-1} f_{i,1}(X)\cdot f_{i,2}(Y) $$ with the polynomials $f_{i,1}(X)$, $f_{i,2}(Y)$ all univariate?