A construction for a class of valuations of the field k(X1,…,Xd,Y) with large value group

Given any algebraically closed field k of characteristic zero and any totally ordered abelian group G of rational rank less than or equal to d, we construct a valuation of the field k(X1,…,Xd,Y) with value group G. In the case of rational rank equal to d this valuation is induced by a formal fractional power series parametrization of a transcendental hypersurface in affine (d+1)-space which is naturally approximated by a sequence of quasi-ordinary hypersurfaces. The value semigroup ν(k[X,Y]∖{0}) is the direct limit of the semigroups associated to these quasi-ordinary hypersurfaces.