Coverage for local/lib/python2.7/site-packages/sage/schemes/curves/constructor.py : 74%

""" 

General curve constructors. 

AUTHORS: 

- William Stein (2005-11-13) 

- David Kohel (2006-01) 

- Grayson Jorgenson (2016-6) 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2005 William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial 

from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing 

from sage.rings.finite_rings.finite_field_constructor import is_FiniteField 

from sage.structure.all import Sequence 

from sage.schemes.affine.affine_space import is_AffineSpace 

from sage.schemes.generic.ambient_space import is_AmbientSpace 

from sage.schemes.generic.algebraic_scheme import is_AlgebraicScheme 

from sage.schemes.projective.projective_space import is_ProjectiveSpace 

from sage.schemes.affine.all import AffineSpace 

from sage.schemes.projective.all import ProjectiveSpace 

from .projective_curve import (ProjectivePlaneCurve, 

ProjectiveCurve, 

ProjectivePlaneCurve_finite_field, 

ProjectivePlaneCurve_prime_finite_field) 

from .affine_curve import (AffinePlaneCurve, 

AffineCurve, 

AffinePlaneCurve_finite_field, 

AffinePlaneCurve_prime_finite_field) 

from sage.schemes.plane_conics.constructor import Conic 

def Curve(F, A=None): 

""" 

Return the plane or space curve defined by ``F``, where 

``F`` can be either a multivariate polynomial, a list or 

tuple of polynomials, or an algebraic scheme. 

If no ambient space is passed in for ``A``, and if ``F`` is not 

an algebraic scheme, a new ambient space is constructed. 

Also not specifying an ambient space will cause the curve to be defined 

in either affine or projective space based on properties of ``F``. In 

particular, if ``F`` contains a nonhomogenous polynomial, the curve is 

affine, and if ``F`` consists of homogenous polynomials, then the curve 

is projective. 

INPUT: 

- ``F`` -- a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme. 

- ``A`` -- (default: None) an ambient space in which to create the curve. 

EXAMPLES: A projective plane curve 

:: 

sage: x,y,z = QQ['x,y,z'].gens() 

sage: C = Curve(x^3 + y^3 + z^3); C 

Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 

sage: C.genus() 

1 

EXAMPLES: Affine plane curves 

:: 

sage: x,y = GF(7)['x,y'].gens() 

sage: C = Curve(y^2 + x^3 + x^10); C 

Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2 

sage: C.genus() 

0 

sage: x, y = QQ['x,y'].gens() 

sage: Curve(x^3 + y^3 + 1) 

Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1 

EXAMPLES: A projective space curve 

:: 

sage: x,y,z,w = QQ['x,y,z,w'].gens() 

sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C 

Projective Curve over Rational Field defined by x^3 + y^3 - z^3 - w^3, x^5 - y*z^4 

sage: C.genus() 

13 

EXAMPLES: An affine space curve 

:: 

sage: x,y,z = QQ['x,y,z'].gens() 

sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C 

Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2 

sage: C.genus() 

47 

EXAMPLES: We can also make non-reduced non-irreducible curves. 

:: 

sage: x,y,z = QQ['x,y,z'].gens() 

sage: Curve((x-y)*(x+y)) 

Projective Conic Curve over Rational Field defined by x^2 - y^2 

sage: Curve((x-y)^2*(x+y)^2) 

Projective Plane Curve over Rational Field defined by x^4 - 2*x^2*y^2 + y^4 

EXAMPLES: A union of curves is a curve. 

:: 

sage: x,y,z = QQ['x,y,z'].gens() 

sage: C = Curve(x^3 + y^3 + z^3) 

sage: D = Curve(x^4 + y^4 + z^4) 

sage: C.union(D) 

Projective Plane Curve over Rational Field defined by 

x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^7 

The intersection is not a curve, though it is a scheme. 

:: 

sage: X = C.intersection(D); X 

Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: 

x^3 + y^3 + z^3, 

x^4 + y^4 + z^4 

Note that the intersection has dimension `0`. 

:: 

sage: X.dimension() 

0 

sage: I = X.defining_ideal(); I 

Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field 

EXAMPLES: In three variables, the defining equation must be 

homogeneous. 

If the parent polynomial ring is in three variables, then the 

defining ideal must be homogeneous. 

:: 

sage: x,y,z = QQ['x,y,z'].gens() 

sage: Curve(x^2+y^2) 

Projective Conic Curve over Rational Field defined by x^2 + y^2 

sage: Curve(x^2+y^2+z) 

Traceback (most recent call last): 

... 

TypeError: x^2 + y^2 + z is not a homogeneous polynomial 

The defining polynomial must always be nonzero:: 

sage: P1.<x,y> = ProjectiveSpace(1,GF(5)) 

sage: Curve(0*x) 

Traceback (most recent call last): 

... 

ValueError: defining polynomial of curve must be nonzero 

:: 

sage: A.<x,y,z> = AffineSpace(QQ, 3) 

sage: C = Curve([y - x^2, z - x^3], A) 

sage: A == C.ambient_space() 

True 

""" 

if not A is None: 

if not isinstance(F, (list, tuple)): 

return Curve([F], A) 

if not is_AmbientSpace(A): 

raise TypeError("A (=%s) must be either an affine or projective space"%A) 

if not all([f.parent() == A.coordinate_ring() for f in F]): 

raise TypeError("F (=%s) must be a list or tuple of polynomials of the coordinate ring of " \ 

"A (=%s)"%(F, A)) 

n = A.dimension_relative() 

if n < 2: 

raise TypeError("A (=%s) must be either an affine or projective space of dimension > 1"%A) 

# there is no dimension check when initializing a plane curve, so check here that F consists 

# of a single nonconstant polynomial 

if n == 2: 

if len(F) != 1 or F[0] == 0 or not is_MPolynomial(F[0]): 

raise TypeError("F (=%s) must consist of a single nonconstant polynomial to define a plane curve"%(F,)) 

if is_AffineSpace(A): 

if n > 2: 

return AffineCurve(A, F) 

k = A.base_ring() 

if is_FiniteField(k): 

if k.is_prime_field(): 

return AffinePlaneCurve_prime_finite_field(A, F[0]) 

return AffinePlaneCurve_finite_field(A, F[0]) 

return AffinePlaneCurve(A, F[0]) 

elif is_ProjectiveSpace(A): 

if not all([f.is_homogeneous() for f in F]): 

raise TypeError("polynomials defining a curve in a projective space must be homogeneous") 

if n > 2: 

return ProjectiveCurve(A, F) 

k = A.base_ring() 

if is_FiniteField(k): 

if k.is_prime_field(): 

return ProjectivePlaneCurve_prime_finite_field(A, F[0]) 

return ProjectivePlaneCurve_finite_field(A, F[0]) 

return ProjectivePlaneCurve(A, F[0]) 

if is_AlgebraicScheme(F): 

return Curve(F.defining_polynomials(), F.ambient_space()) 

if isinstance(F, (list, tuple)): 

if len(F) == 1: 

return Curve(F[0]) 

F = Sequence(F) 

P = F.universe() 

if not is_MPolynomialRing(P): 

raise TypeError("universe of F must be a multivariate polynomial ring") 

for f in F: 

if not f.is_homogeneous(): 

A = AffineSpace(P.ngens(), P.base_ring()) 

A._coordinate_ring = P 

return AffineCurve(A, F) 

A = ProjectiveSpace(P.ngens()-1, P.base_ring()) 

A._coordinate_ring = P 

return ProjectiveCurve(A, F) 

if not is_MPolynomial(F): 

raise TypeError("F (=%s) must be a multivariate polynomial"%F) 

P = F.parent() 

k = F.base_ring() 

if F.parent().ngens() == 2: 

if F == 0: 

raise ValueError("defining polynomial of curve must be nonzero") 

A2 = AffineSpace(2, P.base_ring()) 

A2._coordinate_ring = P 

if is_FiniteField(k): 

if k.is_prime_field(): 

return AffinePlaneCurve_prime_finite_field(A2, F) 

else: 

return AffinePlaneCurve_finite_field(A2, F) 

else: 

return AffinePlaneCurve(A2, F) 

elif F.parent().ngens() == 3: 

if F == 0: 

raise ValueError("defining polynomial of curve must be nonzero") 

P2 = ProjectiveSpace(2, P.base_ring()) 

P2._coordinate_ring = P 

if F.total_degree() == 2 and k.is_field(): 

return Conic(F) 

if is_FiniteField(k): 

if k.is_prime_field(): 

return ProjectivePlaneCurve_prime_finite_field(P2, F) 

else: 

return ProjectivePlaneCurve_finite_field(P2, F) 

else: 

return ProjectivePlaneCurve(P2, F) 

else: 

raise TypeError("Number of variables of F (=%s) must be 2 or 3"%F)