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Homotopy stability of spaces of non-resultant systems of bounded multiplicity with real coefficients

Andrzej Kozlowski, Kohhei Yamaguchi

TL;DR

The paper advances the understanding of spaces $Poly^{d,m}_n(\mathbb{R})$ of $m$-tuples of real monic polynomials of degree $d$ with no common root of multiplicity $n$, focusing on homotopy stability as $d$ grows. Building on prior results for $mn\ge 4$ and $mn=3$ in the complex and real settings, it proves homotopy stability for the real case when $mn=3$ by establishing simplicity up to explicit dimensions for the cases $(m,n)=(3,1)$ and $(1,3)$, and by analyzing the induced maps $i^{d,m}_{n,\mathbb{R}}$, the stabilization maps $s^{d,m}_{n,\mathbb{R}}$, and the jet embedding $j^{d}_n$. The work also provides equivariant (\mathbb{Z}_2) comparisons with the complex setting, and a stable decomposition into wedges of spheres and equivariant half-smash products $D_j$, thereby delivering precise homotopy- and homology-stability statements. Collectively, these results extend the scope of homotopy stability for non-resultant polynomial systems to the challenging $mn=3$ real case and clarify the asymptotic homotopy types via explicit maps and decompositions.

Abstract

We continue our study of the topology of the spaces of $m$ tuples of real polynomials with common degree $d$ and without common roots of multiplicity $n$, and in particular their stability properties with respect to $d$. In an earlier paper we have proved a homotopy stability result and determined the stable homotopy types of such spaces in the case $m n >=4$. In the case $m n= 3$ we could only prove stability in homology. In this paper we prove the corresponding homotopy result for the case $m n =3$.

Homotopy stability of spaces of non-resultant systems of bounded multiplicity with real coefficients

TL;DR

The paper advances the understanding of spaces of -tuples of real monic polynomials of degree with no common root of multiplicity , focusing on homotopy stability as grows. Building on prior results for and in the complex and real settings, it proves homotopy stability for the real case when by establishing simplicity up to explicit dimensions for the cases and , and by analyzing the induced maps , the stabilization maps , and the jet embedding . The work also provides equivariant (\mathbb{Z}_2) comparisons with the complex setting, and a stable decomposition into wedges of spheres and equivariant half-smash products , thereby delivering precise homotopy- and homology-stability statements. Collectively, these results extend the scope of homotopy stability for non-resultant polynomial systems to the challenging real case and clarify the asymptotic homotopy types via explicit maps and decompositions.

Abstract

We continue our study of the topology of the spaces of tuples of real polynomials with common degree and without common roots of multiplicity , and in particular their stability properties with respect to . In an earlier paper we have proved a homotopy stability result and determined the stable homotopy types of such spaces in the case . In the case we could only prove stability in homology. In this paper we prove the corresponding homotopy result for the case .
Paper Structure (13 sections, 21 theorems, 111 equations)

This paper contains 13 sections, 21 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. 1.1 Historical survey.
  3. 1.2 The main results.
  4. 1.3 The organization.
  5. Basic notations and definitions
  6. 2.1 Basic definitions and notations.
  7. 2.2 Spaces of non-resultant systems.
  8. 2.3 The natural maps and stabilization maps.
  9. The case $(m,n)=(3,1)$
  10. 3.1 The space $\hbox{\rm Poly}^{d,3}_1$.
  11. 3.2 Fundamental group actions.
  12. The case $(m,n)=(1,3)$
  13. Proofs of the main results

Key Result

Theorem 1.2

We make the identification $S^2=\mathbb{C}\cup \infty$ and let $(\Omega^2_d\mathbb{C}{\rm P}^1)^{\mathbb{Z}_2}_j$ denote the space of base-point preserving maps $S^2\to \mathbb{C}{\rm P}^1$ of degree $d$ which commute with complex conjugation and have degree $j$ when restricted to the real axis $S^1 $\hbox{\rm (ii)}$ If $j=d-2k$ and $0\leq k\leq d$, the natural inclusion map is a homotopy equival

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2: Br, Se; the case $(m,n)=(2,1)$
  • Theorem 1.3: KY13, Se0; the case $(m,n)=(1,2)$
  • Theorem 1.4: KY13; the case $mn\geq 3$
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7: The case $(m,n)=(3,1)$
  • Theorem 1.8: The case $(m,n)=(1,3)$
  • Corollary 1.9
  • Corollary 1.10
  • ...and 51 more