Abstract
Abstract We prove that projective spaces of Lorentzian and real stable polynomials are homeomorphic to Euclidean balls. This solves a conjecture of June Huh and the author. The proof utilises and refines a connection between the symmetric exclusion process in interacting particle systems and the geometry of polynomials.
Highlights
Over the past two decades there has been a surge of activity in the study of stable, hyperbolic and Lorentzian polynomials
This remarkable space is stratified by the family of all M-convex sets [15] on the discrete simplex. It was proved in [7] that the space PLnd is compact and contractible and conjectured that PLnd is homeomorphic to a closed Euclidean ball. We prove this conjecture here (Theorem 3.4) by utilising and further developing a powerful connection between the symmetric exclusion process (SEP) and the geometry of polynomials, which was discovered and studied in [6]
The proof technique applies to spaces of real stable polynomials, and we prove that these spaces are homeomorphic to closed Euclidean balls
Summary
Over the past two decades there has been a surge of activity in the study of stable, hyperbolic and Lorentzian polynomials. This remarkable space is stratified by the family of all M-convex sets [15] on the discrete simplex It was proved in [7] that the space PLnd is compact and contractible and conjectured that PLnd is homeomorphic to a closed Euclidean ball. SEP contracts the spaces to a point in the interior These results are used, in conjunction with a construction of Galashin, Karp and Lam [9], to prove that various spaces of Lorentzian polynomials are closed Euclidean balls. This refines a result of Nuij [16], who proved that such spaces are contractible. In the final section we identify topics for further studies
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