A note on polynomial maps having fibers of maximal dimension
Volume 166 / 2021
Colloquium Mathematicum 166 (2021), 129-136
MSC: Primary 12D10; Secondary 14E05, 52B11.
DOI: 10.4064/cm8162-8-2020
Published online: 9 March 2021
Abstract
For any two integers $k,n$, $2\leq k\leq n$, let $f:(\mathbb {C}^*)^n\rightarrow \mathbb {C}^k$ be a generic polynomial map with a given Newton polytope. It is known that the points whose fiber under $f$ has codimension 1 form a finite set $C_1(f)$ in $\mathbb {C}^k$. We show that $C_1(f)$ is empty if $k\geq 3$, we classify all Newton polytopes contributing to $C_1(f)\neq \emptyset $ for $k=2$, and we compute $|C_1(f)|$.
