A topological proof of the Shapiro-Shapiro conjecture

• 成果类型: Article
• 署名作者: Levinson, Jake; Purbhoo, Kevin
• 署名单位: Simon Fraser University; University of Waterloo
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01056-y
• 发表日期: 2021
• 页码: 521-578
• 关键词: schubert calculus; real solutions; galois-groups; Lower bounds; REPRESENTATIONS; Transversality; Respect; CURVES

摘要

We prove a generalization of the Shapiro-Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given as an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.