CAT(0) conjecture for the stratified Euclidean metric on polynomial space

Prove that the stratified Euclidean metric on the space _d^{mt} of all monic degree-d complex polynomials up to translation is CAT(0).

Background

The authors introduce a natural stratified Euclidean metric on the polynomial space _d{mt} that reflects the Lyashko–Looijenga stratification. They conjecture that this metric has nonpositive curvature in the CAT(0) sense.

A positive resolution would connect curvature properties of polynomial moduli spaces to those of the dual braid complex and braid groups, potentially providing new geometric tools and implications for long-standing problems in geometric group theory.

References

\begin{conj}\label{conj:cat0} The stratified Euclidean metric on the space $_d{mt}$ of all monic degree-$d$ complex polynomials up to translation is $(0)$. \end{conj}

— Geometric Combinatorics of Polynomials II: Polynomials and Cell Structures  (2410.03047 - Dougherty et al., 2024) in Generalizations and conjectures; Conjecture (label 'conj:cat0')