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For each pair $(m,n)$ of positive integers with $(m,n)\not= (1,1)$ and an arbitrary field $\bf F$ with algebraic closure $\overline{\bf F}$, let $\rm Po^{d,m}_n(\bf F)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \bf F [z]^m$ of $\bf F$-coefficients monic polynomials of the same degree $d$ such that the polynomials $\{f_k(z)\}_{k=1}^m$ have no common root in $\overline{\bf F}$ of multiplicity $\geq n$. These spaces $\rm Po^{d,m}_n(\bf F)$ 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 previous we determined explicitly the homotopy type of this space in the case $\bf F =\Bbb C$. In this paper, we investigate the case $\bf F =\Bbb R$.

We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates of the bounds by using new methods introduced in \cite{Mo3} and used in \cite{KY4}. In addition, in the the last section, we prove a special case ($m=1$) of the conjecture stated in \cite{AKY1} that our spaces are homotopy equivalent to the spaces of algebraic maps.