Spaces of polynomials with roots of bounded multiplicity

The homology of spaces of polynomials with roots of bounded multiplicity

Let P_{k, n}^l be the space consisting of monic complex polynomials f(z) of degree k and such that the number of n-fold roots of f(z) is at most l. In this paper, we determine the integral homology groups of P_{k, n}^l.

Spaces of non-resultant systems of bounded multiplicity determined by a toric variety

The space of non-resultant systems of bounded multiplicity for a toric variety X is a generalization of the space of rational curves on it. In our earlier work we proved a homotopy stability theorem and determined explicitly the homotopy type of this space for the case X = CP^m. In this paper we consider the case of a general non-singular toric variety and prove a homotopy stability theorem generalising the one for CP^m.

The homotopy type of spaces of real resultants with bounded multiplicity

For positive integers $d,m,n\geq 1$ with $(m,n)\not= (1,1)$ and $\Bbb K=\Bbb R$ or $\Bbb C$, let $Q^{d,m}_{n}(\Bbb K)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb K [z]^m$ of $\Bbb K$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_k(z)\}_{k=1}^m$ have no common {\it real} root of multiplicity $\geq n$ (but may have complex common root of any multiplicity). %% These spaces can be regarded as one of generalizations of the spaces defined and studied by Arnold and Vassiliev, and they may be also considered as the real analogues of the spaces st…

The homotopy type of spaces of resultants of bounded multiplicity

For positive integers $m,n, d\geq 1$ with $(m,n)\not= (1,1)$ and a field $\Bbb F$ with its algebraic closure $\overline{\Bbb F}$, let $\text{Poly}^{d,m}_n(\Bbb F)$ denote the space of all $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb F [z]$ of monic polynomials of the same degree $d$ such that polynomials $f_1(z),\cdots ,f_m(z)$ have no common root in $\overline{\Bbb F}$ of multiplicity $\geq n$. These spaces were defined by Farb and Wolfson in \cite{FW} as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. In \cite{FW} they obtained algebraic g…

具有实系数的有界重数非结式系统的空间

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

本文研究实系数情形下，由无重数≥n公共根的d次首一多项式m元组构成的空间$\rm Po^{d,m}_n(\bf F)$的同伦型，该空间是Arnold等人研究的经典空间的推广，此前已确定复系数情形的结果。