Combinatorial Cellular Decompositions for the Space of Complex Coefficient Polynomials

TL;DR

This paper classifies degree n complex coefficient polynomials based on combinatorial patterns from their real and imaginary zero sets, providing explicit results for degree 3 and special polynomial families.

Contribution

It introduces a novel combinatorial classification framework for complex polynomials, extending previous work to singular cases and specific polynomial families.

Findings

Explicit classification for degree 3 polynomials.

Extension of combinatorial methods to singular cases.

Analysis of special polynomial families.

Abstract

We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we work out explicitly this classification for degree 3 polynomials, and other special families of polynomials. This work extends to the singular case similar considerations of Martin, Savitt, and Singer for non-singular basketballs.