The homotopy type of spaces of real resultants with bounded multiplicity
type: Item
instance of: scholarly article
title: The homotopy type of spaces of real resultants with bounded multiplicity
zbMATH Open document ID: 1506.55007
author: Kohhei Yamaguchi
author: Andrzej Kozłowski
published in: Journal of the Mathematical Society of Japan
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1803.02154
review text: Let \(\mathbb{K}\) be the field \(\mathbb{R}\) or \(\mathbb{C}\), and let \(d(\mathbb{K})\) be the dimension of \(\mathbb{K}\) as a real vector space. That is \[ d(\mathbb{K})=\begin{cases} 1 &\text{if }\mathbb{K}=\mathbb{R};\\ 2 &\text{if }\mathbb{K}=\mathbb{C}. \end{cases} \] Let \(Q^{d,m}_n(\mathbb{K})\) be the topological space consisting of \(m\)-tuples \((f_1(z),\ldots,f_m(z))\) of degree-\(d\) monic \(\mathbb{K}\)-polynomials such that \(f_1(z),\ldots,f_m(z)\) have no common real root of multiplicity \(\geq n\). In this paper the authors study the homotopy types of the \(Q^{d,m}_n(\mathbb{K})\)'s and identify them as recognizable spaces in low dimensions. To state the main results, let \(\Omega_{[d]_2}\mathbb{RP}^{N}\) be the connected component of \(\Omega\mathbb{RP}^N\) containing null-homotopic maps for \(d\) even, and let it be the other connected component for \(d\) odd; let \(\jmath^{d,m}_{n,\mathbb{K}}\colon Q^{d,m}_n(\mathbb{K})\to\Omega_{[d]_2}\mathbb{RP}^{d(\mathbb{K})mn-1}\) be the jet map given by \[ \jmath^{d,m}_{n,\mathbb{K}}(f_1(z),\ldots,f_m(z))(t)=\begin{cases} [\textbf{f}_1(t):\cdots:\textbf{f}_m(t)] &\text{if }t\in\mathbb{R}\\ [1:\cdots:1] &\text{if }t=\infty, \end{cases} \] where each \(\textbf{f}_i(z)\) is the \(n\)-tuple \[ \textbf{f}_i(z)=(f_1(z),f_1(z)+f^{(1)}_1(z),f_1(z)+f^{(2)}_1(z),\ldots,f_1(z)+f^{(n-1)}_1(z)), \] and let \(D(d;m,n,\mathbb{K})=(d(\mathbb{K})mn-2)\cdot(\lfloor\frac{d}{n}\rfloor+1)-1\) where \(\lfloor x\rfloor\) denotes the integer part of a real number \(x\). Under the assumption \(d(\mathbb{K})mn\geq4\), the authors prove two main theorems (Theorems 1.8 and 1.11): Theorem 1.8 says that \(\jmath^{d,m}_{n,\mathbb{K}}\) is a homotopy equivalence through dimension \(D(d;m,n,\mathbb{K})\), that is the induced group homomorphism \((\jmath^{d,m}_{n,\mathbb{K}})_*\colon\pi_k(Q^{d,m}_n(\mathbb{K}))\to\pi_k(\Omega_{[d]_2}\mathbb{RP}^{d(\mathbb{K})mn-1})\) is an isomorphism for all \(k\leq D(d;m,n,\mathbb{K})\); Theorem 1.11 …
reviewed by: Tseleung So
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55R80
zbMATH DE Number: 7608347
zbMATH Keywords: jet map
zbMATH Keywords: multiplicity
zbMATH Keywords: resultant
zbMATH Keywords: scanning map
MaRDI profile type: Publication
OpenAlex ID: W2793280661
arXiv ID: 1803.02154
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DOI: 10.2969/JMSJ/79897989
description: scientific article