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

For any field F and positive integers $m,n,d$ with (m,n)= (1,1), Farb and Wolfson defined the certain affine varieties Polyd,mₙ( F) as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others. As a natural generalization of this, for each fan Σ and r-tuple D=(d₁,⋯ ,dᵣ) of positive integers, the current authors defined spaces PolyD,Σₙ( F), where r is the number of one dimensional cones in Σ. These spaces can also be regarded as generalizations of the space Hol^*D(S²,X_Σ) of based rational curves from the Riemann sphere S² to the toric variety X_Σ of degree D, where X_Σ denotes the toric variety (over C) corresponding to the fan Σ. In this paper, we define spaces QD,Σₙ( F) (F= R or C) which are real analogues of PolyD,Σₙ( 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.