Let $\mathbb F$ be a field, find a necessary and sufficient condition on $\mathbb F$ such that the only semi-polynomial maps are the polynomials.

Question

Let $\mathbb F$ be a field.

A map $f : \mathbb F^2 \rightarrow \mathbb F$ is semi-polynomial if for every fixed $y$ the map $ x \rightarrow f(x,y)$ is a polynomial and for every fixed $x$ the map $y \rightarrow f(x,y)$ is a polynomial.

I'm trying to find a necessary and sufficient condition on $\mathbb F$ such that the only semi-polynomial maps are the polynomials. Any ideas would be appreciated.

Answer 1

It follows from this discussion that semi polynomials are polynomials iff $\mathbb{F}$ is not countable infinite.