  We address the question of detecting minimal virtual diagrams with respect to
the number of virtual crossings. This problem is closely connected to the
problem of detecting the minimal number of additional intersection points for a
generic immersion of a singular link in $R^{2}$.
  We tackle this problem by the so-called $\xi$-polynomial whose leading
(lowest) degree naturally estimates the virtual crossing number.
  Several sufficient conditions for minimality together with infinite series of
new examples are given.
  We also state several open questions about $M$-diagrams, which are minimal
according to our sufficient conditions.
