On polynomial invariants of virtual links

The $VA$-polynomial proposed in the author's earlier paper (Acta Appl. Math. 72 (2002), 295-309) for virtual knots and links is considered in this paper. One goal here is to refine the definition of this polynomial to the case of the ring ${\mathbb Z}$ in place of the field ${\mathbb Q}$. Moreover, the approach in the paper mentioned makes it possible to recognize ``long virtual knots'' obtained from equivalent virtual knots by cutting at various points. An invariant of long virtual knots that is based on the same technique as the  $VA$-polynomial is proposed. Some properties of the $VA$-polynomial are established. Furthermore, new invariants of virtual links and long virtual knots are constructed.