In this paper we characterize all possible Hilbert functions of reduced subavrieties of Veronese varieties. In particular, we get an effective description in the case of Veronese curves and surfaces. In these cases we also describe all possible reduced complete intersection subvarieties.
The paper is an introduction to the use of the classical Newton–Puiseux procedure, oriented towards an algorithmic description of it. This procedure allows to obtain polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and is almost self-contained. We illustrate the use of the algorithm first in a completely worked out example of a curve with a point of multiplicity 6, and secondly, in the study of triple points on reduced plane curves.
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