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In this paper we present a method to compute an implicitization of a rational parametrized curve in an affine space over an algebraically closed field. This method is the natural generalization of the resultant method for planar curves. For this purpose we need some normality assumptions on the parametrization of the curve. Furthermore, we provide a test to decide whether a parametrization is normal and if not, we compute a normal parametrization.

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In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms.

We present the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups of that genus. This construction provides an alternative approach to the conjecture on the increasingness of the number of numerical semigroups of each given genus. We elaborate on the number of nodes at each tree depth in the forest and present new conjectures.