Towards effective detection of the bifurcation locus of real polynomial maps

We answer to a problem raised by recent work of Jelonek and Kurdyka: how can one detect by rational arcs the bifurcation locus of a polynomial map ${\text{bR}}^n\to {\text{bR}}^p$ in case $p>1$. We describe an effective estimation of the “nontrivial” part of the bifurcation locus.