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Spaces of non-resultant systems of bounded multiplicity with real coefficients

Andrzej Kozlowski, Kohhei Yamaguchi

TL;DR

This work extends the theory of non-resultant polynomial systems to real coefficients by studying Poly^{d,m}_n(\mathbb{R}), spaces of m-tuples of monic degree-d polynomials with no common root of multiplicity at least n. It develops a Vassiliev-type spectral sequence to analyze the real discriminant and proves an Atiyah–Jones–Segal-type stabilization: the natural real map i^{d,m}_{n,\mathbb{R}}: Poly^{d,m}_n(\mathbb{R}) \to (\Omega^2_d\mathbb{C}P^{mn-1})^{\mathbb{Z}_2} is a homotopy equivalence through dimension $D(d;m,n)$ (when $mn\ge4$, and a homology equivalence through that range when $mn=3$). The paper also obtains a stable decomposition Poly^{d,m}_n(\mathbb{R}) \simeq_s Poly^{\lfloor d/2\rfloor,m}_n(\mathbb{C}) \vee B^{d,m}_n \vee Q^{d,m}_n(\mathbb{R}) with an explicit description in terms of wedge sums of suspensions of the spaces D_j, connecting real and complex pictures via scanning and stabilization maps. This yields homology stability and, in many cases, homotopy stability for the real spaces, and clarifies the role of the Z_2-action in the real-analytic setting. The appendix treats the edge case (m,n)=(1,2), illustrating braid-structure components and showing stable behavior does not hold uniformly there, in contrast to the broader regime.

Abstract

For each pair $(m,n)$ of positive integers with $(m,n)\not= (1,1)$ and an arbitrary field $\bf F$ with algebraic closure $\overline{\bf F}$, let $\rm Po^{d,m}_n(\bf F)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \bf F [z]^m$ of $\bf F$-coefficients monic polynomials of the same degree $d$ such that the polynomials $\{f_k(z)\}_{k=1}^m$ have no common root in $\overline{\bf F}$ of multiplicity $\geq n$. These spaces $\rm Po^{d,m}_n(\bf F)$ were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In previous we determined explicitly the homotopy type of this space in the case $\bf F =\Bbb C$. In this paper, we investigate the case $\bf F =\Bbb R$.

Spaces of non-resultant systems of bounded multiplicity with real coefficients

TL;DR

This work extends the theory of non-resultant polynomial systems to real coefficients by studying Poly^{d,m}_n(\mathbb{R}), spaces of m-tuples of monic degree-d polynomials with no common root of multiplicity at least n. It develops a Vassiliev-type spectral sequence to analyze the real discriminant and proves an Atiyah–Jones–Segal-type stabilization: the natural real map i^{d,m}_{n,\mathbb{R}}: Poly^{d,m}_n(\mathbb{R}) \to (\Omega^2_d\mathbb{C}P^{mn-1})^{\mathbb{Z}_2} is a homotopy equivalence through dimension (when , and a homology equivalence through that range when ). The paper also obtains a stable decomposition Poly^{d,m}_n(\mathbb{R}) \simeq_s Poly^{\lfloor d/2\rfloor,m}_n(\mathbb{C}) \vee B^{d,m}_n \vee Q^{d,m}_n(\mathbb{R}) with an explicit description in terms of wedge sums of suspensions of the spaces D_j, connecting real and complex pictures via scanning and stabilization maps. This yields homology stability and, in many cases, homotopy stability for the real spaces, and clarifies the role of the Z_2-action in the real-analytic setting. The appendix treats the edge case (m,n)=(1,2), illustrating braid-structure components and showing stable behavior does not hold uniformly there, in contrast to the broader regime.

Abstract

For each pair of positive integers with and an arbitrary field with algebraic closure , let denote the space of -tuples of -coefficients monic polynomials of the same degree such that the polynomials have no common root in of multiplicity . These spaces were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In previous we determined explicitly the homotopy type of this space in the case . In this paper, we investigate the case .
Paper Structure (13 sections, 42 theorems, 225 equations)

This paper contains 13 sections, 42 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. The motivation.
  3. Basic definitions and the main results
  4. Basic definitions and notations.
  5. Some known results.
  6. The main results.
  7. The spaces $(\Omega^2_d\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_2}$ and $\hbox{\rm Poly}^{d,m}_n(\mathbb{R};\mathbf{\rm H}_+)$
  8. The Vassiliev spectral sequence
  9. Loop products and stabilization maps
  10. The homology stability theorem
  11. Configuration spaces and scanning maps
  12. The homotopy type of $\hbox{\rm Poly}^{d,m}_n(\mathbb{R})$
  13. Appendix: The case $(m,n)=(1,2)$

Key Result

Theorem 1.1

Let $m,n,d\geq 1$ be positive integers such that $(m,n)\not= (1,1)$ with $d\geq n$,If $d<n$, the space $\hbox{\rm Poly}^{d,m}_n(\mathbb{R})$ is contractible by (eq: contractible), so assume that $d\geq n$. and let $D(d;m,n)$ denote the positive integer defined by where $\lfloor x\rfloor$ denotes the integer part of a real number $x$. $\hbox{\rm (i)}$ The natural map $($defined by $($eq: mapjR$))$

Theorems & Definitions (106)

  • Theorem 1.1: Theorems \ref{['thm: KY13']}, \ref{['thm: KY13; stable homotopy type']} and Corollary \ref{['crl: KY13; stable homotopy type']}
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: KY8
  • Theorem 2.6: KY10, Va, Y1
  • Theorem 2.7
  • Corollary 2.8
  • ...and 96 more