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Combinatorial Cellular Decompositions for the Space of Complex Coefficient Polynomials

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.