The $K$-Theory of Versal Flags and Cohomological Invariants of Degree 3

Let G be a split semisimple linear algebraic group over a field and let X be a generic twisted flag variety of G . Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring K_0(X) in terms of generators and relations in the case G=G^{sc}/\mu_2> is of Dynkin type A or C (here G^{sc} is the simply-connected cover of G ); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.