A note on polynomial maps having fibers of maximal dimension

For any two integers $k,n$, $2\le k\le n$, let $f:({\mathbb{C}}^{\ast }{)}^n\to {\mathbb{C}}^k$ be a generic polynomial map with given Newton polytopes. It is known that points, whose fiber under $f$ has codimension one, form a finite set $C_1(f)$ in ${\mathbb{C}}^k$. For maps $f$ above, we show that $C_1(f)$ is empty if $k\ge 3$, we classify all Newton polytopes contributing to $C_1(f)\mathrm{\ne }\mathrm{\varnothing }$ for $k=2$, and we compute $\mathrm{\mid }{C}_1(f)\mathrm{\mid }$.