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We address the question of detecting minimal virtual diagrams with respect tothe number of virtual crossings. This problem is closely connected to theproblem of detecting the minimal number of additional intersection points for ageneric 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 ofnew examples are given.  We also state several open questions about $M$-diagrams, which are minimalaccording to our sufficient conditions.

ON VIRTUAL CROSSING NUMBER ESTIMATES FOR VIRTUAL LINKS