# Signatures of monic polynomials

## TL;DR

This paper introduces a novel combinatorial framework linking monic polynomials to planar forests called signatures, demonstrating their realizability, and exploring their topological properties and implications for braid group cohomology.

## Contribution

It establishes that all combinatorially possible signatures are realizable and describes the topology of polynomial spaces associated with these signatures.

## Key findings

- All signatures are realizable by polynomials.
- Spaces of polynomials with a fixed signature are contractible.
- A finite cell complex for braid group cohomology is constructed.

## Abstract

To a univariate monic polynomial is attached a special planar forest that is called the picture of the polynomial. Isotopy classes of pictures are called signatures. All combinatorially possible signatures are realized and spaces of polynomials realizing a given signature are contractible. A finite cell complex for the cohomology of the braid groups is obtained.