Non-smooth points set of fibres of a semialgebraic mapping

Abstract

For a semialgebraic mapping between semialgebraic sets, we consider the set of points at which the fibre is not smooth. In this paper we discuss whether the singular set is itself semialgebraic, when it has codimension bigger than or equal to 2 in the domain of $f$ and whether the mapping is semialgebraically trivial along the smooth part of the fibre, giving several examples which show optimality of those results. In addition, we give an example of a polynomial function $f$ such that even the $(a_f)$ condition in the weak sense fails in a neighbourhood of a smooth fibre, but $f$ is semialgebraically trivial along it.