Let $f(x, y), g(x, y) \in \mathbb{C}[x, y]$. Consider the image $S$ of the map

$(f, g): \mathbb{C}^2 \rightarrow \mathbb{C}^2, (z_1, z_2) \mapsto (f(z_1, z_2), g(z_1, z_2)).$


If for an integer $d \in \mathbb{N}$ the set $S$ contains a set $A_1 \times A_2, A_1, A_2 \subset \mathbb{C}$ being such that the minimum of the cardinalities of $|A_1|, |A_2|$ is $d$, one can say that "$S$ contains a product set of width $d$". 

If $S$ contains product sets of arbitrarily large width, then one can observe that it follows from the <a hyperlink="http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBQQFjAA&url=http%3A%2F%2Fwww.tau.ac.il%2F~nogaa%2FPDFS%2Fnull2.pdf&rct=j&q=Combinatorial%20Nullstellensatz&ei=m7UdToKKBc_2sga6oJCyDQ&usg=AFQjCNHwCZrLJCZ_dVub9yeTxNCmNJ1cZQ&cad=rja">Combinatorial Nullstellensatz</a> that the polynomials $f(x, y)$ and $g(x, y)$ are algebraically independent over $\mathbb{C}$. 

I wonder whether it is known if the converse is true (that is, if $f(x, y)$ and $g(x, y)$ are algebraically independent over $\mathbb{C}$, then $S$ contains product sets of arbitrarily large width)?