Geometric Combinatorics of Polynomials I: The Case of a Single Polynomial

TL;DR

This paper introduces a new geometric object associated with complex polynomials that encodes various algebraic, geometric, and combinatorial properties, providing a unified framework for analyzing polynomial roots and related structures.

Contribution

It constructs a canonical compact planar 2-complex with a CAT(0) metric that encodes multiple combinatorial and algebraic invariants of a polynomial with distinct roots.

Findings

The 2-complex is a locally CAT(0) metric space with Euclidean properties away from critical points.

Various combinatorial data can be extracted from the complex using metric graphs like cacti and banyans.

The construction provides a unified geometric framework for understanding polynomial invariants.

Abstract

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial $p$ with $d$ distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally CAT(0) metric that is locally Euclidean away from a finite set of interior points indexed by the critical points of $p$, and each of its 2-cells is a metric rectangle. From this planar rectangular 2-complex one can use metric graphs known as metric cacti and metric banyans to read off several pieces of combinatorial data: a chain in the partition lattice, a cyclic factorization of a d-cycle, a real noncrossing partition (also known as a primitive d-major), and the monodromy permutations for the polynomial. This article is the first in a series.