If $f: \mathbb{R}^2 \to \mathbb{R}$ is polynomial in each argument, $f$ is polynomial

Question

Given $f: \mathbb{R}^2 \to \mathbb{R}$ such that for all real $y$, $t \mapsto f(t,y)$ is a polynomial function and for all real $x$, $t \mapsto f(x,t)$ is a polynomial function, is it necessarily true that $(x,y) \mapsto f(x,y)$ is a polynomial function?