Spaces of nonresultant systems of bounded multiplicity with real coefficients

Kozlowski, Andrzej; Yamaguchi, Kohhei

doi:10.2140/agt.2026.26.2589

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Spaces of nonresultant systems of bounded multiplicity with real coefficients

Andrzej Kozlowski and Kohhei Yamaguchi

Algebraic & Geometric Topology 26 (2026) 2589–2633

DOI: 10.2140/agt.2026.26.2589

Abstract

For each pair $(m, n)$ of positive integers with $(m, n) \neq (1, 1)$ and an arbitrary field $𝔽$ with algebraic closure $\bar{𝔽}$ , let ${Poly}_{n}^{d, m} (𝔽)$ denote the space of $m$ -tuples $(f_{1} (z), \dots, f_{m} (z)) \in 𝔽 {[z]}^{m}$ of monic polynomials with coefficients in $𝔽$ of the same degree $d$ such that the polynomials ${f_{k} (z)}_{k = 1}^{m}$ have no common root in $\bar{𝔽}$ of multiplicity $\geq n$ . These spaces ${Poly}_{n}^{d, m} (𝔽)$ 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 a previous paper we determined explicitly the homotopy type of this space when $𝔽 = ℂ$ . Here we investigate this space when $𝔽 = ℝ$ .

Keywords

homotopy type, multiplicity, resultant, jet map, scanning map, configuration space, spaces of nonresultant systems, homotopy stability, Vassiliev spectral sequence

Mathematical Subject Classification

Primary: 55P15

Secondary: 55P35, 55R80

References

Publication

Received: 3 December 2024

Revised: 26 June 2025

Accepted: 26 July 2025

Published: 1 September 2026

Authors

Andrzej Kozlowski
Institute of Applied Mathematics and Mechanics University of Warsaw Warsaw Poland

Kohhei Yamaguchi
Department of Mathematics University of Electro-Communications Tokyo Japan

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Volume 26, issue 7 (2026)