
D.5.4 paraplanecurves_lib
 Library:
 paraplanecurves.lib
 Purpose:
 Rational parametrization of rational plane curves
 Authors:
 J. Boehm, j.boehm at mx.unisaarland.de
W. Decker, decker at mathematik.unikl.de>
S. Laplagne, slaplagn at dm.uba.ar
F. Seelisch, seelisch at mathematik.unikl.de
 Overview:
 Suppose C = {f(x,y,z)=0} is a rational plane curve, where f is homogeneous
of degree n with coefficients in Q and absolutely irreducible (these
conditions are checked automatically.)
After a first step, realized by a projective automorphism in the procedure
adjointIdeal, C satisfies:
 C does not have singularities at infinity z=0.
 C does not contain the point (0:1:0) (that is, the dehomogenization of f
with respect to z is monic as a polynomial in y).
Considering C in the chart z<>0, the algorithm regards x as transcendental
and y as algebraic and computes an integral basis in C(x)[y] of the integral
closure of C[x] in C(x,y) using the normalization algorithm from
normal_lib: see integralbasis_lib. In a future edition of the
library, also van Hoeij's algorithm for computing the integral basis will
be available.
From the integral basis, the adjoint ideal is obtained by linear algebra.
Alternatively, the algorithm starts with a local analysis of the singular
locus of C. Then, for each primary component of the singular locus which
does not correspond to ordinary multiple points or cusps, the integral
basis algorithm is applied separately. The ordinary multiple points and
cusps, in turn, are addressed by a straightforward direct algorithm. The
adjoint ideal is obtained by intersecting all ideals obtained locally.
The local variant of the algorithm is used by default.
The linear system corresponding to the adjoint ideal maps the curve
birationally to a rational normal curve in P^(n2).
Iterating the anticanonical map, the algorithm projects the rational normal
curve to PP1 for n odd resp. to a conic C2 in PP2 for n even.
In case n is even, the algorithm tests whether there is a rational point on
C2 and if so gives a parametrization of C2 which is defined over Q. Otherwise,
the parametrization given is defined over a quadratic field extension of Q.
By inverting the birational map of C to PP1 resp. to C2, a parametrization
of C is obtained (defined over Q or the quadratic field extension).
 References:
 Janko Boehm: Parametrisierung rationaler Kurven, Diploma Thesis,
http://www.math.unisb.de/ag/schreyer/jb/diplom%20janko%20boehm.pdf
Theo de Jong: An algorithm for computing the integral closure,
Journal of Symbolic Computation 26 (3) (1998), p. 273277
GertMartin Greuel, Santiago Laplagne, Frank Seelisch: Normalization of Rings,
Journal of Symbolic Computation 9 (2010), p. 887901
Mark van Hoeij: An Algorithm for Computing an Integral Basis in an Algebraic
Function Field, Journal of Symbolic Computation 18 (1994), p. 353363,
http://www.math.fsu.edu/~hoeij/papers/comments/jsc1994.html
Procedures:
