Implicitization of de Jonquières parametrizations. (English) Zbl 1311.14014

Hassanzadeh, Seyed Hamid; Simis, Aron

J. Commut. Algebra 6, No. 2, 149-172 (2014).

A de Jonquières parametrization \({\mathcal F}\) of a hypersurface is derived from a given Cremona map of \(n\)-dimensional projective space onto itself. Such parametrizations are specials forms of rational parametrizations. They can also be seen as generalizations of monoid parametrizations.
The goal of this paper is to derive the implicit equation \(F=0\) for the hypersurface from the data involved in a de Jonquières parametrization \({\mathcal F}\); or, in any case, as much information about the defining polynomial \(F\) as possible. So, for instance, a formula for the degree of \(F\) is given. Furthermore, the relation between the present parametrization and monoid parametrizations is exhibited.
An indication of which aspects of the treatment can be carried out algorithmically, and how, would definitely increase the value of this approach.

Reviewer: Franz Winkler (Linz)

MSC:

14E05	Rational and birational maps
13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14E07	Birational automorphisms, Cremona group and generalizations
13D02	Syzygies, resolutions, complexes and commutative rings
14Q99	Computational aspects in algebraic geometry
13H15	Multiplicity theory and related topics
13D45	Local cohomology and commutative rings

Keywords:

de Jonquières parametrization; implicitization

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