  Generalizing earlier results about the set of idempotents in a Banach
algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce
constructions of nice connecting paths in the connected components of the set
of elements in a Banach algebra, or of self-adjoint elements in a
$C^*$-algebra, that satisfy a given polynomial equation, without multiple
roots. In particular, we will prove that in the Banach algebra case every such
non-central element lies on a complex line, all of whose points satisfy the
given equation. We also formulate open questions.
