Geodesic
On the Parametrization of an Algebraic Curve
Matematičeskie zametki, Tome 106 (2019) no. 6, pp. 837-847

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At present, a plane algebraic curve can be parametrized in the following two cases: if its genus is equal to 0 or 1 and if it has a large group of birational automorphisms. Here we propose a new polyhedron method (involving a polyhedron called a Hadamard polyhedron by the author), which allows us to divide the space $\mathbb R^2$ or $\mathbb C^2$ into pieces in each of which the polynomial specifying the curve is sufficiently well approximated by its truncated polynomial, which often defines the parametrized curve. This approximate parametrization in a piece can be refined by means of the Newton method. Thus, an arbitrarily exact piecewise parametrization of the original curve can be obtained.
Keywords: algebraic curve, genus of a curve, piecewise parametrization, Hadamard polyhedron, Newton method. 
@article{MZM_2019_106_6_a3,
     author = {A. D. Bruno},
     title = {On the {Parametrization} of an {Algebraic} {Curve}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {837--847},
     publisher = {mathdoc},
     volume = {106},
     number = {6},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_6_a3/}
}
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A. D. Bruno. On the Parametrization of an Algebraic Curve. Matematičeskie zametki, Tome 106 (2019) no. 6, pp. 837-847. http://geodesic.mathdoc.fr/item/MZM_2019_106_6_a3/