In a Banach algebra, whenever you can connect two idempotents by an idempotent-valued continuous path, you can as well assume that the latter is polynomial [Esterle 1983]. In the finite-dimensional case, you can also take this polynomial to have degree less than or equal to 3 and the bound 3 is optimal [Tremon 1985, Esterle-G. 2004]. It can be shown that there is a uniform bound for von Neumann algebras, but it remains to compute its precise value. We will review all the known estimates and report on some recent attempts to determine this bound for some other classes of algebras.