Coverage for local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/constructor.py : 66%
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""" Hyperelliptic curve constructor """ from __future__ import absolute_import
#***************************************************************************** # Copyright (C) 2006 David Kohel <kohel@maths.usyd.edu> # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #*****************************************************************************
from sage.schemes.projective.projective_space import ProjectiveSpace
from .hyperelliptic_generic import HyperellipticCurve_generic from .hyperelliptic_finite_field import HyperellipticCurve_finite_field from .hyperelliptic_rational_field import HyperellipticCurve_rational_field from .hyperelliptic_padic_field import HyperellipticCurve_padic_field from .hyperelliptic_g2_generic import HyperellipticCurve_g2_generic from .hyperelliptic_g2_finite_field import HyperellipticCurve_g2_finite_field from .hyperelliptic_g2_rational_field import HyperellipticCurve_g2_rational_field from .hyperelliptic_g2_padic_field import HyperellipticCurve_g2_padic_field
from sage.rings.padics.all import is_pAdicField
from sage.rings.rational_field import is_RationalField from sage.rings.finite_rings.finite_field_constructor import is_FiniteField from sage.rings.polynomial.polynomial_element import is_Polynomial
def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True): r""" Returns the hyperelliptic curve `y^2 + h y = f`, for univariate polynomials `h` and `f`. If `h` is not given, then it defaults to 0.
INPUT:
- ``f`` - univariate polynomial
- ``h`` - optional univariate polynomial
- ``names`` (default: ``["x","y"]``) - names for the coordinate functions
- ``check_squarefree`` (default: ``True``) - test if the input defines a hyperelliptic curve when f is homogenized to degree `2g+2` and h to degree `g+1` for some g.
.. WARNING::
When setting ``check_squarefree=False`` or using a base ring that is not a field, the output curves are not to be trusted. For example, the output of ``is_singular`` is always ``False``, without this being properly tested in that case.
.. NOTE::
The words "hyperelliptic curve" are normally only used for curves of genus at least two, but this class allows more general smooth double covers of the projective line (conics and elliptic curves), even though the class is not meant for those and some outputs may be incorrect.
EXAMPLES:
Basic examples::
sage: R.<x> = QQ[] sage: HyperellipticCurve(x^5 + x + 1) Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1 sage: HyperellipticCurve(x^19 + x + 1, x-2) Hyperelliptic Curve over Rational Field defined by y^2 + (x - 2)*y = x^19 + x + 1
sage: k.<a> = GF(9); R.<x> = k[] sage: HyperellipticCurve(x^3 + x - 1, x+a) Hyperelliptic Curve over Finite Field in a of size 3^2 defined by y^2 + (x + a)*y = x^3 + x + 2
Characteristic two::
sage: P.<x> = GF(8,'a')[] sage: HyperellipticCurve(x^7+1, x) Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + x*y = x^7 + 1 sage: HyperellipticCurve(x^8+x^7+1, x^4+1) Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + (x^4 + 1)*y = x^8 + x^7 + 1
sage: HyperellipticCurve(x^8+1, x) Traceback (most recent call last): ... ValueError: Not a hyperelliptic curve: highly singular at infinity.
sage: HyperellipticCurve(x^8+x^7+1, x^4) Traceback (most recent call last): ... ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
sage: F.<t> = PowerSeriesRing(FiniteField(2)) sage: P.<x> = PolynomialRing(FractionField(F)) sage: HyperellipticCurve(x^5+t, x) Hyperelliptic Curve over Laurent Series Ring in t over Finite Field of size 2 defined by y^2 + x*y = x^5 + t
We can change the names of the variables in the output::
sage: k.<a> = GF(9); R.<x> = k[] sage: HyperellipticCurve(x^3 + x - 1, x+a, names=['X','Y']) Hyperelliptic Curve over Finite Field in a of size 3^2 defined by Y^2 + (X + a)*Y = X^3 + X + 2
This class also allows curves of genus zero or one, which are strictly speaking not hyperelliptic::
sage: P.<x> = QQ[] sage: HyperellipticCurve(x^2+1) Hyperelliptic Curve over Rational Field defined by y^2 = x^2 + 1 sage: HyperellipticCurve(x^4-1) Hyperelliptic Curve over Rational Field defined by y^2 = x^4 - 1 sage: HyperellipticCurve(x^3+2*x+2) Hyperelliptic Curve over Rational Field defined by y^2 = x^3 + 2*x + 2
Double roots::
sage: P.<x> = GF(7)[] sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1)) Traceback (most recent call last): ... ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1), check_squarefree=False) Hyperelliptic Curve over Finite Field of size 7 defined by y^2 = x^12 + 5*x^10 + 4*x^9 + x^8 + 3*x^7 + 3*x^6 + 2*x^4 + 3*x^3 + 6*x^2 + 4*x + 3
The input for a (smooth) hyperelliptic curve of genus `g` should not contain polynomials of degree greater than `2g+2`. In the following example, the hyperelliptic curve has genus 2 and there exists a model `y^2 = F` of degree 6, so the model `y^2 + yh = f` of degree 200 is not allowed.::
sage: P.<x> = QQ[] sage: h = x^100 sage: F = x^6+1 sage: f = F-h^2/4 sage: HyperellipticCurve(f, h) Traceback (most recent call last): ... ValueError: Not a hyperelliptic curve: highly singular at infinity.
sage: HyperellipticCurve(F) Hyperelliptic Curve over Rational Field defined by y^2 = x^6 + 1
An example with a singularity over an inseparable extension of the base field::
sage: F.<t> = GF(5)[] sage: P.<x> = F[] sage: HyperellipticCurve(x^5+t) Traceback (most recent call last): ... ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
Input with integer coefficients creates objects with the integers as base ring, but only checks smoothness over `\QQ`, not over Spec(`\ZZ`). In other words, it is checked that the discriminant is non-zero, but it is not checked whether the discriminant is a unit in `\ZZ^*`.::
sage: P.<x> = ZZ[] sage: HyperellipticCurve(3*x^7+6*x+6) Hyperelliptic Curve over Integer Ring defined by y^2 = 3*x^7 + 6*x + 6
TESTS:
Check that `f` can be a constant (see :trac:`15516`)::
sage: R.<u> = PolynomialRing(Rationals()) sage: HyperellipticCurve(-12, u^4 + 7) Hyperelliptic Curve over Rational Field defined by y^2 + (x^4 + 7)*y = -12
""" # F is the discriminant; use this for the type check # rather than f and h, one of which might be constant. raise TypeError("Arguments f (= %s) and h (= %s) must be polynomials" % (f, h)) else: # Assuming we are working over a field, this checks that after # resolving the singularity at infinity, we get a smooth double cover # of P^1. # characteristic 2 raise ValueError("In characteristic 2, argument h (= %s) must be non-zero."%h) "highly singular at infinity.") else: # characteristic not 2 "highly singular at infinity.") except (AttributeError, NotImplementedError, TypeError): try: smooth = should_be_coprime[0].resultant(should_be_coprime[1])!=0 except (AttributeError, NotImplementedError, TypeError): raise NotImplementedError("Cannot determine whether " \ "polynomials %s have a common root. Use " \ "check_squarefree=False to skip this check." % \ should_be_coprime) "singularity in the provided affine patch.") else: else: else: else: else: |