A construction for a class of valuations of the field [formula omitted] with large value group

Abstract

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 ( X 1 , … , X d , 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.