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gh-36271: `sage.{dynamics,schemes}`: Modularization fixes, docstring cosmetics, update `# needs` <!-- ^^^^^ Please provide a concise, informative and self-explanatory title. Don't put issue numbers in there, do this in the PR body below. For example, instead of "Fixes #1234" use "Introduce new method to calculate 1+1" --> <!-- Describe your changes here in detail --> <!-- Why is this change required? What problem does it solve? --> <!-- If this PR resolves an open issue, please link to it here. For example "Fixes #12345". --> - Part of: #29705 - Cherry-picked from: #35095 <!-- If your change requires a documentation PR, please link it appropriately. --> ### 📝 Checklist <!-- Put an `x` in all the boxes that apply. --> <!-- If your change requires a documentation PR, please link it appropriately --> <!-- If you're unsure about any of these, don't hesitate to ask. We're here to help! --> <!-- Feel free to remove irrelevant items. --> - [x] The title is concise, informative, and self-explanatory. - [ ] The description explains in detail what this PR is about. - [x] I have linked a relevant issue or discussion. - [ ] I have created tests covering the changes. - [ ] I have updated the documentation accordingly. ### ⌛ Dependencies <!-- List all open PRs that this PR logically depends on - #12345: short description why this is a dependency - #34567: ... --> <!-- If you're unsure about any of these, don't hesitate to ask. We're here to help! --> URL: #36271 Reported by: Matthias Köppe Reviewer(s): David Coudert, Matthias Köppe
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‎src/sage/dynamics/arithmetic_dynamics/affine_ds.py‎

Lines changed: 42 additions & 26 deletions
Original file line numberDiff line numberDiff line change
@@ -162,8 +162,9 @@ class DynamicalSystem_affine(SchemeMorphism_polynomial_affine_space,
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If you pass in quotient ring elements, they are reduced::
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sage: # needs sage.libs.singular
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sage: A.<x,y,z> = AffineSpace(QQ, 3)
166-
sage: X = A.subscheme([x-y])
167+
sage: X = A.subscheme([x - y])
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sage: u,v,w = X.coordinate_ring().gens()
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sage: DynamicalSystem_affine([u, v, u+v], domain=X)
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Dynamical System of Closed subscheme of Affine Space of dimension 3
@@ -174,9 +175,10 @@ class DynamicalSystem_affine(SchemeMorphism_polynomial_affine_space,
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::
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sage: # needs sage.libs.singular
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sage: R.<t> = PolynomialRing(QQ)
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sage: A.<x,y,z> = AffineSpace(R, 3)
179-
sage: X = A.subscheme(x^2-y^2)
181+
sage: X = A.subscheme(x^2 - y^2)
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sage: H = End(X)
181183
sage: f = H([x^2/(t*y), t*y^2, x*z])
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sage: DynamicalSystem_affine(f)
@@ -188,8 +190,8 @@ class DynamicalSystem_affine(SchemeMorphism_polynomial_affine_space,
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::
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191-
sage: x = var('x')
192-
sage: DynamicalSystem_affine(x^2+1)
193+
sage: x = var('x') # needs sage.symbolic
194+
sage: DynamicalSystem_affine(x^2 + 1) # needs sage.symbolic
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Traceback (most recent call last):
194196
...
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TypeError: symbolic ring cannot be the base ring
@@ -397,10 +399,11 @@ def homogenize(self, n):
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::
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sage: # needs sage.rings.number_field
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sage: R.<a> = PolynomialRing(QQbar)
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sage: A.<x,y> = AffineSpace(R, 2)
402-
sage: f = DynamicalSystem_affine([QQbar(sqrt(2))*x*y, a*x^2])
403-
sage: f.homogenize(2)
405+
sage: f = DynamicalSystem_affine([QQbar(sqrt(2))*x*y, a*x^2]) # needs sage.symbolic
406+
sage: f.homogenize(2) # needs sage.symbolic
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Dynamical System of Projective Space of dimension 2 over Univariate
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Polynomial Ring in a over Algebraic Field
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Defn: Defined on coordinates by sending (x0 : x1 : x2) to
@@ -453,7 +456,7 @@ def dynatomic_polynomial(self, period):
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sage: A.<x> = AffineSpace(ZZ, 1)
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sage: f = DynamicalSystem_affine([(x^2+1)/x])
456-
sage: f.dynatomic_polynomial(4)
459+
sage: f.dynatomic_polynomial(4) # needs sage.libs.pari
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2*x^12 + 18*x^10 + 57*x^8 + 79*x^6 + 48*x^4 + 12*x^2 + 1
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::
@@ -507,8 +510,8 @@ def dynatomic_polynomial(self, period):
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::
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sage: A.<x> = AffineSpace(CC,1)
510-
sage: F = DynamicalSystem_affine([1/2*x^2 + CC(sqrt(3))])
511-
sage: F.dynatomic_polynomial([1,1])
513+
sage: F = DynamicalSystem_affine([1/2*x^2 + CC(sqrt(3))]) # needs sage.symbolic
514+
sage: F.dynatomic_polynomial([1,1]) # needs sage.symbolic
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(0.125000000000000*x^4 + 0.366025403784439*x^2 + 1.50000000000000)/(0.500000000000000*x^2 - x + 1.73205080756888)
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TESTS::
@@ -770,7 +773,7 @@ def multiplier(self, P, n, check=True):
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sage: P.<x> = AffineSpace(CC, 1)
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sage: f = DynamicalSystem_affine([x^2 + 1/2])
773-
sage: f.multiplier(P([0.5 + 0.5*I]), 1)
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sage: f.multiplier(P([0.5 + 0.5*I]), 1) # needs sage.symbolic
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[1.00000000000000 + 1.00000000000000*I]
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::
@@ -784,7 +787,7 @@ def multiplier(self, P, n, check=True):
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::
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sage: P.<x,y> = AffineSpace(QQ, 2)
787-
sage: X = P.subscheme([x^2-y^2])
790+
sage: X = P.subscheme([x^2 - y^2])
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sage: f = DynamicalSystem_affine([x^2, y^2], domain=X)
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sage: f.multiplier(X([1, 1]), 1)
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[2 0]
@@ -823,7 +826,7 @@ def conjugate(self, M):
823826
EXAMPLES::
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sage: A.<t> = AffineSpace(QQ, 1)
826-
sage: f = DynamicalSystem_affine([t^2+1])
829+
sage: f = DynamicalSystem_affine([t^2 + 1])
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sage: f.conjugate(matrix([[1,2], [0,1]]))
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Dynamical System of Affine Space of dimension 1 over Rational Field
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Defn: Defined on coordinates by sending (t) to
@@ -832,18 +835,20 @@ def conjugate(self, M):
832835
::
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sage: A.<x,y> = AffineSpace(ZZ,2)
835-
sage: f = DynamicalSystem_affine([x^3+y^3,y^2])
838+
sage: f = DynamicalSystem_affine([x^3 + y^3, y^2])
836839
sage: f.conjugate(matrix([[1,2,3], [0,1,2], [0,0,1]]))
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Dynamical System of Affine Space of dimension 2 over Integer Ring
838841
Defn: Defined on coordinates by sending (x, y) to
839-
(x^3 + 6*x^2*y + 12*x*y^2 + 9*y^3 + 9*x^2 + 36*x*y + 40*y^2 + 27*x + 58*y + 28, y^2 + 4*y + 2)
842+
(x^3 + 6*x^2*y + 12*x*y^2 + 9*y^3
843+
+ 9*x^2 + 36*x*y + 40*y^2 + 27*x + 58*y + 28, y^2 + 4*y + 2)
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::
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sage: # needs sage.rings.number_field
843848
sage: R.<x> = PolynomialRing(QQ)
844-
sage: K.<i> = NumberField(x^2+1)
849+
sage: K.<i> = NumberField(x^2 + 1)
845850
sage: A.<x> = AffineSpace(ZZ,1)
846-
sage: f = DynamicalSystem_affine([x^3+2*x^2+3])
851+
sage: f = DynamicalSystem_affine([x^3 + 2*x^2 + 3])
847852
sage: f.conjugate(matrix([[i,i], [0,-i]]))
848853
Dynamical System of Affine Space of dimension 1 over Integer Ring
849854
Defn: Defined on coordinates by sending (x) to
@@ -860,6 +865,7 @@ def degree(self):
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EXAMPLES::
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868+
sage: # needs sage.rings.number_field
863869
sage: R.<c> = QuadraticField(7)
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sage: A.<x,y,z> = AffineSpace(R, 3)
865871
sage: f = DynamicalSystem_affine([x^2 + y^5 + c, x^11, z^19])
@@ -911,16 +917,18 @@ def weil_restriction(self):
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EXAMPLES::
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920+
sage: # needs sage.rings.number_field
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sage: K.<v> = QuadraticField(5)
915922
sage: A.<x,y> = AffineSpace(K, 2)
916-
sage: f = DynamicalSystem_affine([x^2-y^2, y^2])
923+
sage: f = DynamicalSystem_affine([x^2 - y^2, y^2])
917924
sage: f.weil_restriction()
918925
Dynamical System of Affine Space of dimension 4 over Rational Field
919926
Defn: Defined on coordinates by sending (z0, z1, z2, z3) to
920927
(z0^2 + 5*z1^2 - z2^2 - 5*z3^2, 2*z0*z1 - 2*z2*z3, z2^2 + 5*z3^2, 2*z2*z3)
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::
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931+
sage: # needs sage.rings.number_field
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sage: K.<v> = QuadraticField(5)
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sage: PS.<x,y> = AffineSpace(K, 2)
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sage: f = DynamicalSystem_affine([x, y])
@@ -940,13 +948,14 @@ def reduce_base_field(self):
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The base field of the map could be strictly larger than
941949
the field where all of the coefficients are defined. This function
942950
reduces the base field to the minimal possible. This can be done when
943-
the base ring is a number field, QQbar, a finite field, or algebraic
951+
the base ring is a number field, ``QQbar``, a finite field, or algebraic
944952
closure of a finite field.
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OUTPUT: A dynamical system
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EXAMPLES::
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958+
sage: # needs sage.rings.finite_rings
950959
sage: K.<t> = GF(5^2)
951960
sage: A.<x,y> = AffineSpace(K, 2)
952961
sage: f = DynamicalSystem_affine([x^2 + 3*y^2, 3*y^2])
@@ -957,15 +966,20 @@ def reduce_base_field(self):
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::
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969+
sage: # needs sage.rings.number_field sage.symbolic
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sage: A.<x,y> = AffineSpace(QQbar, 2)
961-
sage: f = DynamicalSystem_affine([x^2 + QQbar(sqrt(3))*y^2, QQbar(sqrt(-1))*y^2])
971+
sage: f = DynamicalSystem_affine([x^2 + QQbar(sqrt(3))*y^2,
972+
....: QQbar(sqrt(-1))*y^2])
962973
sage: f.reduce_base_field()
963-
Dynamical System of Affine Space of dimension 2 over Number Field in a with defining polynomial y^4 - y^2 + 1 with a = -0.866025403784439? + 0.50000000000000000?*I
974+
Dynamical System of Affine Space of dimension 2 over Number Field in a
975+
with defining polynomial y^4 - y^2 + 1
976+
with a = -0.866025403784439? + 0.50000000000000000?*I
964977
Defn: Defined on coordinates by sending (x, y) to
965978
(x^2 + (a^3 - 2*a)*y^2, (a^3)*y^2)
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::
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sage: # needs sage.rings.number_field
969983
sage: K.<v> = CyclotomicField(5)
970984
sage: A.<x,y> = AffineSpace(K, 2)
971985
sage: f = DynamicalSystem_affine([(3*x^2 + y) / (5*x), (y^2+1) / (x+y)])
@@ -1001,6 +1015,7 @@ def orbit_structure(self, P):
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10021016
::
10031017
1018+
sage: # needs sage.rings.finite_rings
10041019
sage: A.<x,y,z> = AffineSpace(GF(49, 't'), 3)
10051020
sage: f = DynamicalSystem_affine([x^2 - z, x - y + z, y^2 - x^2])
10061021
sage: f.orbit_structure(A(1, 1, 2))
@@ -1028,23 +1043,24 @@ def cyclegraph(self):
10281043
EXAMPLES::
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10301045
sage: P.<x,y> = AffineSpace(GF(5), 2)
1031-
sage: f = DynamicalSystem_affine([x^2-y, x*y+1])
1032-
sage: f.cyclegraph()
1046+
sage: f = DynamicalSystem_affine([x^2 - y, x*y + 1])
1047+
sage: f.cyclegraph() # needs sage.graphs
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Looped digraph on 25 vertices
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::
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1052+
sage: # needs sage.rings.finite_rings
10371053
sage: P.<x> = AffineSpace(GF(3^3, 't'), 1)
1038-
sage: f = DynamicalSystem_affine([x^2-1])
1039-
sage: f.cyclegraph()
1054+
sage: f = DynamicalSystem_affine([x^2 - 1])
1055+
sage: f.cyclegraph() # needs sage.graphs
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Looped digraph on 27 vertices
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10421058
::
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sage: P.<x,y> = AffineSpace(GF(7), 2)
1045-
sage: X = P.subscheme(x-y)
1061+
sage: X = P.subscheme(x - y)
10461062
sage: f = DynamicalSystem_affine([x^2, y^2], domain=X)
1047-
sage: f.cyclegraph()
1063+
sage: f.cyclegraph() # needs sage.graphs
10481064
Looped digraph on 7 vertices
10491065
"""
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V = []

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