Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fields of Parametrization and Optimal Affine Reparametrization of
  Rational Curves   In this paper we present three related results on the subject of fields of
parametrization. Let C be a rational curve over a field of characteristic zero.
Let K be a field finitely generated over Q, such that it is a field of
definition of C but not a field of parametrization. It is known that there are
quadratic extensions of K that parametrize C. First, we prove that there are
infinitely many quadratic extensions of K that are fields of parametrization of
C.
  As a consequence, we prove that the witness variety, that appear in the
context of the parametric Weil's descente method, is always a special curve
related to algebraic extensions, called hypercircle. It is possible that the
witness variety is not a hypercircle for the given extension, but for an
alternative one.
  We use these two facts to present an algorithm to solve the following optimal
reparametrization problem. Given a birational parametrization f(t) of a curve
C, compute the affine reparametrization at+b such f(at+b) has coefficients over
a field as small as possible. The main advantage of this algorithm is that it
does not need to compute any rational point on the curve.

33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th