A note on polynomial maps having fibers of maximal dimension

TL;DR

This paper investigates the structure of polynomial maps from complex tori to complex space, showing that certain fibers are empty in higher dimensions, classifying cases with non-empty fibers, and calculating their number.

Contribution

It provides a classification of Newton polytopes related to fibers of maximal dimension in polynomial maps from complex tori, extending known results to higher dimensions.

Findings

Fibers of codimension one are empty for maps with target dimension at least 3.

Classifies Newton polytopes that lead to non-empty fibers in the case of two-dimensional targets.

Calculates the exact number of such fibers for specific polynomial maps.

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 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 $k\geq 3$, we classify all Newton polytopes contributing to $C_1(f)\neq \emptyset$ for $k=2$, and we compute $|C_1(f)|$.