On the braid monodromy group of a polynomial in one variable

TL;DR

This paper proves that the braid monodromy group of a polynomial with n-1 distinct critical values is isomorphic to the braid group on n strands, providing insight into the algebraic structure of polynomial mappings.

Contribution

It establishes a condition under which the braid monodromy group of a polynomial equals the full braid group, advancing understanding of polynomial monodromy groups.

Findings

Braid monodromy group equals the braid group when polynomial has n-1 distinct critical values.

Provides a criterion linking critical values to the structure of the monodromy group.

Enhances the theoretical understanding of polynomial monodromy in complex algebraic geometry.

Abstract

It is proved that the braid monodromy group $\Gamma_{P(z)}$ of a polynomial $P(z)\in\mathbb C[z]$, $\deg P(z)=n$, is the braid group $\text{Br}_n$ if the polynomial $P(z)$ has $n-1$ distinct critical values.