  We show that the space $\tilde{A}_{d}(m,n)$ consisting of all real projective
classes of $(n+1)$-tuples of real coefficients homogeneous polynomials of
degree $d$ in $(m+1)$ variables, without common real roots except zero, has the
same homology as the space $ \Map(\RP^m,\Bbb \RP^n)$ of continuous maps from
the $m$-dimensional real projective space $\RP^m$ into the $n$ real dimensional
projective space $\RP^n$ up to dimension %in dimensions smaller than
$(n-m)(d+1)-1$. This considerably improves the main result of \cite{AKY1}.
