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

TL;DR

This paper introduces and analyzes real analogues of polynomial spaces associated with toric varieties, proving homotopy stability and calculating stability dimensions, thus extending classical results in real singularity theory.

Contribution

It defines real polynomial spaces related to toric varieties and establishes their homotopy stability with explicit stability dimensions, generalizing prior complex and real singularity results.

Findings

Homotopy stability holds for the defined spaces.

Explicit stability dimensions are computed.

Generalizes classical spaces studied by Arnold and Vassiliev.

Abstract

For any field $\Bbb F$ and positive integers $m,n,d$ with $(m,n)\not= (1,1)$, Farb and Wolfson defined the certain affine varieties ${\rm Poly}^{d,m}_n(\Bbb F)$ as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others. As a natural generalization of this, for each fan $\Sigma$ and $r$-tuple $D=(d_1,\cdots ,d_r)$ of positive integers, the current authors defined spaces ${\rm Poly}^{D,\Sigma}_n(\Bbb F)$, where $r$ is the number of one dimensional cones in $\Sigma$. These spaces can also be regarded as generalizations of the space ${\rm Hol}^*_D(S^2,X_\Sigma)$ of based rational curves from the Riemann sphere $S^2$ to the toric variety $X_\Sigma$ of degree $D$, where $X_\Sigma$ denotes the toric variety (over $\Bbb C$) corresponding to the fan $\Sigma$. In this paper, we define spaces ${\rm Q}^{D,\Sigma}_n(\Bbb F)$ ($\Bbb F=\Bbb R$ or $\Bbb C$) which are real analogues of ${\rm Poly}^{D,\Sigma}_n(\Bbb F)$ and which can be viewed as a generalizations of spaces considered by Arnold, Vassiliev and others in the context of real singularity theory. We prove that homotopy stability holds for these spaces and compute the stability dimensions explicitly.