{{{id=1| PS.=ProjectiveSpace(QQ,1) PS /// Projective Space of dimension 1 over Rational Field }}} {{{id=2| PS.coordinate_ring() /// Multivariate Polynomial Ring in x, y over Rational Field }}} {{{id=3| PS.dimension() /// 1 }}} {{{id=4| R.=PolynomialRing(QQ) PC.=ProjectiveSpace(R,1) #point in projective space Q=PC([2+c,4]) Q.codomain() Q.domain() #endomorphism of Projective Space H=Hom(PC,PC) f=H([x^2+c*y^2,y^2]) f /// Scheme endomorphism of Projective Space of dimension 1 over Univariate Polynomial Ring in c over Rational Field Defn: Defined on coordinates by sending (x : y) to (x^2 + c*y^2 : y^2) }}} {{{id=5| f(PC(0,1)) /// (c : 1) }}} {{{id=6| f.nth_iterate(PC(0,1),3) /// (c^4 + 2*c^3 + c^2 + c : 1) }}} {{{id=7| P.=ProjectiveSpace(QQ,1) H=Hom(P,P) f=H([x^2+y^2,y^2]) f.orbit(P(1,1),5) /// [(1 : 1), (1 : 1), (1 : 1), (1 : 1), (1 : 1), (1 : 1)] }}} {{{id=8| P.point([0,0],check=False) /// (0 : 0) }}} {{{id=9| f.primes_of_bad_reduction() /// [2] }}} {{{id=10| P.=ProjectiveSpace(QQ,1) H=Hom(P,P) f=H([x^2,x*y]) f /// Scheme endomorphism of Projective Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x : y) to (x^2 : x*y) }}} {{{id=11| f.orbit(P(0,1),5,check=False) /// [(0 : 1), (0 : 0), (0 : 0), (0 : 0), (0 : 0), (0 : 0)] }}} {{{id=12| P.=ProjectiveSpace(QQ,2) Q=P(3,2,1) Q2=Q.dehomogenize(0) /// }}} {{{id=13| Q2.codomain() /// Affine Space of dimension 2 over Rational Field }}} {{{id=14| Q.scale_by(2) Q /// (6 : 4 : 2) }}} {{{id=15| Q.scale_by(1/10) Q /// (3/5 : 2/5 : 1/5) }}} {{{id=16| Q.clear_denominators() Q /// (3 : 2 : 1) }}} {{{id=18| Q[0] /// 3 }}} {{{id=19| f[0] /// x^2 }}} {{{id=20| P. = ProjectiveSpace(ZZ,1) H = Hom(P,P) f = H([x^2+y^2,y^2]) g=f.conjugate(matrix([[1,CC.0],[0,1]])) g /// Scheme endomorphism of Projective Space of dimension 1 over Multivariate Polynomial Ring in x, y over Complex Field with 53 bits of precision Defn: Defined on coordinates by sending (x : y) to (x^2 + 2.00000000000000*I*x*y + (-1.00000000000000*I)*y^2 : y^2) }}} {{{id=22| f /// Scheme endomorphism of Projective Space of dimension 1 over Integer Ring Defn: Defined on coordinates by sending (x : y) to (x^2 + y^2 : y^2) }}} {{{id=23| R.=GF(5^2) P. = ProjectiveSpace(R,1) H = Hom(P,P) f = H([x^2+y^2,y^2]) /// }}} {{{id=24| f.cyclegraph().show(figsize=10) /// }}} {{{id=25| f.orbit_structure(P(t,1)) /// [1, 2] }}} {{{id=26| P.=ProjectiveSpace(QQ,1) H=Hom(P,P) f=H([x^2-y^2,y^2]) Q=P(25/73, 23/117);Q /// (2925/1679 : 1) }}} {{{id=27| Q.global_height() /// 7.98104975966596 }}} {{{id=28| f.global_height() /// 0 }}} {{{id=29| f.canonical_height(Q) /// 7.6979234162573314972063192355 }}} {{{id=30| f.canonical_height(f(Q)) /// 15.395846832514662994412638471 }}} {{{id=31| f.canonical_height(Q,error_bound=.0001) /// 7.6979234162573314972063192355 }}} {{{id=32| Q.canonical_height(f,error_bound=.0001) /// 7.6979234162573314972063192355 }}} {{{id=33| f.possible_periods() /// [1, 2] }}} {{{id=34| f.rational_preperiodic_graph().show() /// }}} {{{id=35| %time P.=ProjectiveSpace(QQ,1) H=Hom(P,P) f=H([x^2-29/16*y^2,y^2]) f.rational_preperiodic_graph().show() /// CPU time: 0.38 s, Wall time: 0.40 s }}} {{{id=36| %time P.=ProjectiveSpace(QQ,2) H=Hom(P,P) f=H([x^2-29/16*z^2,y^2-21/16*z^2,z^2]) f.rational_preperiodic_points() /// [(5/4 : 5/4 : 1), (-1/4 : 5/4 : 1), (-5/4 : -5/4 : 1), (0 : 1 : 0), (-1 : 1 : 0), (-3/4 : 7/4 : 1), (3/4 : -3/4 : 1), (-7/4 : 1/4 : 1), (-1/4 : -7/4 : 1), (3/4 : 7/4 : 1), (-3/4 : 3/4 : 1), (5/4 : 1/4 : 1), (1/4 : 7/4 : 1), (1/4 : -5/4 : 1), (1/4 : -3/4 : 1), (5/4 : 3/4 : 1), (-3/4 : 5/4 : 1), (7/4 : 1/4 : 1), (5/4 : -7/4 : 1), (7/4 : 5/4 : 1), (3/4 : 3/4 : 1), (-7/4 : 3/4 : 1), (-5/4 : -1/4 : 1), (-7/4 : -5/4 : 1), (-3/4 : -5/4 : 1), (-3/4 : -1/4 : 1), (5/4 : -3/4 : 1), (7/4 : 3/4 : 1), (-3/4 : -7/4 : 1), (7/4 : -1/4 : 1), (-7/4 : 7/4 : 1), (-1/4 : 1/4 : 1), (7/4 : 7/4 : 1), (-7/4 : -1/4 : 1), (-5/4 : -7/4 : 1), (1/4 : 1/4 : 1), (1/4 : -1/4 : 1), (3/4 : 5/4 : 1), (-5/4 : 3/4 : 1), (1/4 : 3/4 : 1), (-1/4 : 3/4 : 1), (7/4 : -5/4 : 1), (5/4 : -1/4 : 1), (1 : 0 : 0), (-5/4 : 1/4 : 1), (1/4 : 5/4 : 1), (-3/4 : 1/4 : 1), (5/4 : 7/4 : 1), (-1/4 : -5/4 : 1), (5/4 : -5/4 : 1), (-5/4 : 7/4 : 1), (-7/4 : -7/4 : 1), (-7/4 : -3/4 : 1), (-1/4 : -3/4 : 1), (7/4 : -7/4 : 1), (3/4 : -5/4 : 1), (3/4 : 1/4 : 1), (3/4 : -7/4 : 1), (1/4 : -7/4 : 1), (-1/4 : 7/4 : 1), (7/4 : -3/4 : 1), (-5/4 : -3/4 : 1), (-1/4 : -1/4 : 1), (-5/4 : 5/4 : 1), (-7/4 : 5/4 : 1), (1 : 1 : 0), (3/4 : -1/4 : 1), (-3/4 : -3/4 : 1)] CPU time: 1.65 s, Wall time: 1.75 s }}} {{{id=38| A.=AffineSpace(QQ,1) A.projective_embedding() /// Scheme morphism: From: Affine Space of dimension 1 over Rational Field To: Projective Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x) to (x : 1) }}} {{{id=37| P.=ProjectiveSpace(QQ,2) E=P.subscheme(y^2*z-x^3+z^3) H=Hom(E,E) f=H([x^2+y^2,y^2,z^2]) /// }}} {{{id=39| f(E(0,1,1)) /// Traceback (most recent call last): File "", line 1, in File "_sage_input_121.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZihFKDAsMSwxKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/var/folders/z2/hs3p8tq16zjb564d79jjlbc80000gp/T/tmpF3XngX/___code___.py", line 3, in exec compile(u'f(E(_sage_const_0 ,_sage_const_1 ,_sage_const_1 ))' + '\n', '', 'single') File "", line 1, in File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/generic/scheme.py", line 294, in __call__ return self.point(args) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/generic/scheme.py", line 370, in point return self.point_homset() (v, check=check) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/generic/homset.py", line 263, in __call__ return Set_generic.__call__(self, *args, **kwds) File "parent.pyx", line 1011, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:8392) File "coerce_maps.pyx", line 100, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_with_args (sage/structure/coerce_maps.c:4278) File "coerce_maps.pyx", line 90, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_with_args (sage/structure/coerce_maps.c:4089) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/generic/homset.py", line 455, in _element_constructor_ return self.codomain()._point(self, v, **kwds) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/generic/algebraic_scheme.py", line 583, in _point return self.__A._point(*args, **kwds) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/projective/projective_space.py", line 827, in _point return SchemeMorphism_point_projective_field(*args, **kwds) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/projective/projective_point.py", line 1042, in __init__ X.extended_codomain()._check_satisfies_equations(v) File "/Applications/sage-5.12/local/lib/python2.7/site-packages/sage/schemes/generic/algebraic_scheme.py", line 967, in _check_satisfies_equations raise TypeError, "Coordinates %s do not define a point on %s"%(coords,self) TypeError: Coordinates [0, 1, 1] do not define a point on Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: -x^3 + y^2*z + z^3 }}} {{{id=40| E /// Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: -x^3 + y^2*z + z^3 }}} {{{id=41| EllipticCurve('37b').category() /// Category of schemes over Rational Field }}} {{{id=42| /// }}}