Signatures of Monic Polynomials

Let P : ℂ → ℂ $$P:\mathbb {C}\to \mathbb {C}$$ be a monic polynomial map of degree d ≥ 1. We call the inverse image of the union of the real and imaginary axes the geometric picture of the polynomial P. The geometric picture of a monic polynomial is a piecewise smooth planar graph. Smooth isotopy classes relative to the 4d asymptotic ends at infinity of geometric pictures are called signatures. The set of signatures Σd of monic degree-d polynomials is finite. We give a combinatorial characterization of the set of signatures Σd and prove that the space of monic polynomials of given signature is contractible. This construction leads to a real semi-algebraic cell-decomposition Pol d = ⋃ σ ∈ Σ d { P ∣ σ ( P ) = σ } $$\displaystyle \mathrm {Pol}_d=\bigcup _{\sigma \in \Sigma _d} \{P \mid \sigma (P)=\sigma \} $$ of the space Pold of monic polynomials of degree d. In this cell-decomposition the classical discriminant locus Δd appears as a union of cells. The complement of the classical discriminant B d := Pold ∖ Δd is a union of cells. The face operators of this cell-decomposition of the space B d are explicitly given. Since B d is a classifying space for the braid group, we obtain a finite complex that computes the group cohomology of the braid group with integral coefficients. The picture of the polynomial P is in fact a union of leaves of the pair of orthogonal foliations of the quadratic differential dP 2. Clearly, our inspiration on this work came from William Thurston’s work.