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Merged
merged 33 commits into from
Apr 6, 2023
Merged

sage.schemes: Reformat doctests, add # optional annotations #35314

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c09a90a
sage.schemes: Add # optional - sage.rings.{number_field,finite_rings,…
mkoeppe Mar 20, 2023
e5e29f7
src/sage/schemes/generic/algebraic_scheme.py: Fix up # optional
mkoeppe Mar 20, 2023
5e7d3f0
sage.schemes: More # optional
mkoeppe Mar 21, 2023
4f53f6e
sage.schemes.curves: Add # optional
mkoeppe Mar 22, 2023
7f6ea46
sage.schemes: Fix up # optional
mkoeppe Mar 22, 2023
b778689
sage.schemes: More cosmetic changes to doctests
mkoeppe Mar 22, 2023
919ed50
sage.schemes: More # optional and cosmetic doctest changes
mkoeppe Mar 22, 2023
47bca76
sage.schemes: More # optional and cosmetic doctest changes (fixup)
mkoeppe Mar 23, 2023
d97da6f
sage.schemes: More cosmetic doctest changes
mkoeppe Mar 23, 2023
0eaf7b1
sage.schemes: More cosmetic doctest changes
mkoeppe Mar 23, 2023
6d21c17
sage.schemes: More # optional and cosmetic doctest changes
mkoeppe Mar 24, 2023
fdb625e
sage.schemes: More # optional and cosmetic doctest changes (fixup)
mkoeppe Mar 24, 2023
a658137
sage.schemes: More cosmetic doctest changes
mkoeppe Mar 24, 2023
2507c8f
sage.schemes: Even more cosmetic doctest changes
mkoeppe Mar 24, 2023
9c7ae71
sage.schemes: Yet more cosmetic doctest changes
mkoeppe Mar 25, 2023
3781322
sage.schemes: Yet more cosmetic doctest changes (fixup)
mkoeppe Mar 25, 2023
61a7aeb
src/sage/schemes/toric/points.py: Fix typo
mkoeppe Mar 25, 2023
7bc90fc
sage.schemes: Some more cosmetic doctest changes
mkoeppe Mar 25, 2023
fbfbd28
src/sage/schemes/affine/affine_homset.py: Add missing space
mkoeppe Mar 29, 2023
0f935d0
src/sage/schemes/elliptic_curves: Revise indentation of multiline doc…
mkoeppe Mar 29, 2023
20bb0f6
src/sage/schemes/curves: Revise indentation of multiline doctest output
mkoeppe Mar 29, 2023
50cc089
src/sage/schemes/elliptic_curves: Revise indentation of multiline doc…
mkoeppe Mar 29, 2023
9cc0865
src/sage/schemes/elliptic_curves/{ell_number_field,height,hom_composi…
mkoeppe Mar 29, 2023
e934a4c
src/sage/schemes: Revise indentation of multiline doctest output
mkoeppe Mar 29, 2023
7e48c64
src/sage/schemes/elliptic_curves/ell_number_field.py: Fix RST markup
mkoeppe Mar 29, 2023
7bc2783
src/sage/schemes: More revisions to the indentation of multiline doct…
mkoeppe Mar 29, 2023
ad35577
src/sage/schemes/berkovich/berkovich_space.py: Fix up # optional
mkoeppe Mar 30, 2023
2564510
src/sage/schemes/elliptic_curves/hom_frobenius.py: Mark sage.doctest:…
mkoeppe Mar 30, 2023
fcdd127
src/sage/schemes/elliptic_curves/isogeny_class.py: Mark sage.doctest:…
mkoeppe Mar 30, 2023
7f9cb68
src/sage/schemes/elliptic_curves/isogeny_small_degree.py: Add # optio…
mkoeppe Mar 30, 2023
3b34a78
src/sage/schemes/overview.py: Fix up # optional
mkoeppe Mar 30, 2023
a9588d1
src/sage/schemes/{affine,projective}: Cosmetic changes to doctests
mkoeppe Mar 30, 2023
912e520
src/sage/schemes/elliptic_curves/ell_generic.py: Fix up doctest
mkoeppe Apr 2, 2023
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32 changes: 18 additions & 14 deletions src/sage/schemes/affine/affine_homset.py
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Original file line number Diff line number Diff line change
Expand Up @@ -210,14 +210,14 @@ def points(self, **kwds):
EXAMPLES: The bug reported at #11526 is fixed::

sage: A2 = AffineSpace(ZZ, 2)
sage: F = GF(3)
sage: A2(F).points()
sage: F = GF(3) # optional - sage.rings.finite_rings
sage: A2(F).points() # optional - sage.rings.finite_rings
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]

::

sage: A.<x,y> = ZZ[]
sage: I = A.ideal(x^2-y^2-1)
sage: I = A.ideal(x^2 - y^2 - 1)
sage: V = AffineSpace(ZZ, 2)
sage: X = V.subscheme(I)
sage: M = X(ZZ)
Expand All @@ -227,9 +227,9 @@ def points(self, **kwds):
::

sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: A.<x,y> = AffineSpace(K, 2)
sage: len(A(K).points(bound=2))
sage: K.<v> = NumberField(u^2 + 3) # optional - sage.rings.number_field
sage: A.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field
sage: len(A(K).points(bound=2)) # optional - sage.rings.number_field
1849

::
Expand All @@ -244,16 +244,20 @@ def points(self, **kwds):
sage: A.<x,y> = AffineSpace(CC, 2)
sage: E = A.subscheme([y^3 - x^3 - x^2, x*y])
sage: E(A.base_ring()).points()
verbose 0 (...: affine_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
verbose 0 (...: affine_homset.py, points)
Warning: computations in the numerical fields are inexact;points
may be computed partially or incorrectly.
[(-1.00000000000000, 0.000000000000000),
(0.000000000000000, 0.000000000000000)]
(0.000000000000000, 0.000000000000000)]

::

sage: A.<x1,x2> = AffineSpace(CDF, 2)
sage: E = A.subscheme([x1^2 + x2^2 + x1*x2, x1 + x2])
sage: E(A.base_ring()).points()
verbose 0 (...: affine_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
verbose 0 (...: affine_homset.py, points)
Warning: computations in the numerical fields are inexact;points
may be computed partially or incorrectly.
[(0.0, 0.0)]
"""
from sage.schemes.affine.affine_space import is_AffineSpace
Expand Down Expand Up @@ -388,14 +392,14 @@ def numerical_points(self, F=None, **kwds):

EXAMPLES::

sage: K.<v> = QuadraticField(3)
sage: A.<x,y> = AffineSpace(K, 2)
sage: X = A.subscheme([x^3 - v^2*y, y - v*x^2 + 3])
sage: L = X(K).numerical_points(F=RR); L # abs tol 1e-14
sage: K.<v> = QuadraticField(3) # optional - sage.rings.number_field
sage: A.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field
sage: X = A.subscheme([x^3 - v^2*y, y - v*x^2 + 3]) # optional - sage.rings.number_field
sage: L = X(K).numerical_points(F=RR); L # abs tol 1e-14 # optional - sage.rings.number_field
[(-1.18738247880014, -0.558021142104134),
(1.57693558184861, 1.30713548084184),
(4.80659931965815, 37.0162574656220)]
sage: L[0].codomain()
sage: L[0].codomain() # optional - sage.rings.number_field
Affine Space of dimension 2 over Real Field with 53 bits of precision

::
Expand Down
431 changes: 209 additions & 222 deletions src/sage/schemes/affine/affine_morphism.py
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62 changes: 31 additions & 31 deletions src/sage/schemes/affine/affine_point.py
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Original file line number Diff line number Diff line change
Expand Up @@ -194,9 +194,9 @@ def global_height(self, prec=None):
::

sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2+5)
sage: A = AffineSpace(k, 2, 'z')
sage: A([3, 5*w+1]).global_height(prec=100)
sage: k.<w> = NumberField(x^2 + 5) # optional - sage.rings.number_field
sage: A = AffineSpace(k, 2, 'z') # optional - sage.rings.number_field
sage: A([3, 5*w + 1]).global_height(prec=100) # optional - sage.rings.number_field
2.4181409534757389986565376694

.. TODO::
Expand Down Expand Up @@ -289,24 +289,24 @@ def weil_restriction(self):

EXAMPLES::

sage: A.<x,y,z> = AffineSpace(GF(5^3, 't'), 3)
sage: X = A.subscheme([y^2-x*z, z^2+y])
sage: Y = X.weil_restriction()
sage: P = X([1, -1, 1])
sage: Q = P.weil_restriction();Q
sage: A.<x,y,z> = AffineSpace(GF(5^3, 't'), 3) # optional - sage.rings.finite_rings
sage: X = A.subscheme([y^2 - x*z, z^2 + y]) # optional - sage.rings.finite_rings
sage: Y = X.weil_restriction() # optional - sage.rings.finite_rings
sage: P = X([1, -1, 1]) # optional - sage.rings.finite_rings
sage: Q = P.weil_restriction();Q # optional - sage.rings.finite_rings
(1, 0, 0, 4, 0, 0, 1, 0, 0)
sage: Q.codomain() == Y
sage: Q.codomain() == Y # optional - sage.rings.finite_rings
True

::

sage: R.<x> = QQ[]
sage: K.<w> = NumberField(x^5-2)
sage: R.<x> = K[]
sage: L.<v> = K.extension(x^2+w)
sage: A.<x,y> = AffineSpace(L, 2)
sage: P = A([w^3-v,1+w+w*v])
sage: P.weil_restriction()
sage: K.<w> = NumberField(x^5 - 2) # optional - sage.rings.number_field
sage: R.<x> = K[] # optional - sage.rings.number_field
sage: L.<v> = K.extension(x^2 + w) # optional - sage.rings.number_field
sage: A.<x,y> = AffineSpace(L, 2) # optional - sage.rings.number_field
sage: P = A([w^3 - v, 1 + w + w*v]) # optional - sage.rings.number_field
sage: P.weil_restriction() # optional - sage.rings.number_field
(w^3, -1, w + 1, w)
"""
L = self.codomain().base_ring()
Expand Down Expand Up @@ -356,14 +356,14 @@ def intersection_multiplicity(self, X):

EXAMPLES::

sage: A.<x,y> = AffineSpace(GF(17), 2)
sage: X = A.subscheme([y^2 - x^3 + 2*x^2 - x])
sage: Y = A.subscheme([y - 2*x + 2])
sage: Q1 = Y([1,0])
sage: Q1.intersection_multiplicity(X)
sage: A.<x,y> = AffineSpace(GF(17), 2) # optional - sage.rings.finite_rings
sage: X = A.subscheme([y^2 - x^3 + 2*x^2 - x]) # optional - sage.rings.finite_rings
sage: Y = A.subscheme([y - 2*x + 2]) # optional - sage.rings.finite_rings
sage: Q1 = Y([1,0]) # optional - sage.rings.finite_rings
sage: Q1.intersection_multiplicity(X) # optional - sage.rings.finite_rings
2
sage: Q2 = X([4,6])
sage: Q2.intersection_multiplicity(Y)
sage: Q2 = X([4,6]) # optional - sage.rings.finite_rings
sage: Q2.intersection_multiplicity(Y) # optional - sage.rings.finite_rings
1

::
Expand Down Expand Up @@ -416,27 +416,27 @@ def __hash__(self):

EXAMPLES::

sage: P.<x,y,z> = AffineSpace(GF(5), 3)
sage: hash(P(2, 1, 2))
sage: P.<x,y,z> = AffineSpace(GF(5), 3) # optional - sage.rings.finite_rings
sage: hash(P(2, 1, 2)) # optional - sage.rings.finite_rings
57

::

sage: P.<x,y,z> = AffineSpace(GF(7), 3)
sage: X = P.subscheme(x^2-y^2)
sage: hash(X(1, 1, 2))
sage: P.<x,y,z> = AffineSpace(GF(7), 3) # optional - sage.rings.finite_rings
sage: X = P.subscheme(x^2 - y^2) # optional - sage.rings.finite_rings
sage: hash(X(1, 1, 2)) # optional - sage.rings.finite_rings
106

::

sage: P.<x,y> = AffineSpace(GF(13), 2)
sage: hash(P(3, 4))
sage: P.<x,y> = AffineSpace(GF(13), 2) # optional - sage.rings.finite_rings
sage: hash(P(3, 4)) # optional - sage.rings.finite_rings
55

::

sage: P.<x,y> = AffineSpace(GF(13^3, 't'), 2)
sage: hash(P(3, 4))
sage: P.<x,y> = AffineSpace(GF(13^3, 't'), 2) # optional - sage.rings.finite_rings
sage: hash(P(3, 4)) # optional - sage.rings.finite_rings
8791
"""
p = self.codomain().base_ring().order()
Expand Down
91 changes: 46 additions & 45 deletions src/sage/schemes/affine/affine_rational_point.py
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Original file line number Diff line number Diff line change
Expand Up @@ -17,16 +17,16 @@

sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
sage: A.<x,y,z> = AffineSpace(3, QQ)
sage: S = A.subscheme([2*x-3*y])
sage: S = A.subscheme([2*x - 3*y])
sage: enum_affine_rational_field(S, 2)
[(0, 0, -2), (0, 0, -1), (0, 0, -1/2), (0, 0, 0),
(0, 0, 1/2), (0, 0, 1), (0, 0, 2)]

Affine over a finite field::

sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
sage: A.<w,x,y,z> = AffineSpace(4, GF(2))
sage: enum_affine_finite_field(A(GF(2)))
sage: A.<w,x,y,z> = AffineSpace(4, GF(2)) # optional - sage.rings.finite_rings
sage: enum_affine_finite_field(A(GF(2))) # optional - sage.rings.finite_rings
[(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 0),
(0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 0, 0, 1),
(1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0),
Expand Down Expand Up @@ -77,15 +77,15 @@ def enum_affine_rational_field(X, B):
sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
sage: enum_affine_rational_field(A(QQ), 1)
[(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]

::

sage: A.<w,x,y,z> = AffineSpace(4, QQ)
sage: S = A.subscheme([x^2-y*z+1, w^3+z+y^2])
sage: S = A.subscheme([x^2 - y*z + 1, w^3 + z + y^2])
sage: enum_affine_rational_field(S(QQ), 1)
[(0, 0, -1, -1)]
sage: enum_affine_rational_field(S(QQ), 2)
Expand All @@ -94,11 +94,11 @@ def enum_affine_rational_field(X, B):
::

sage: A.<x,y> = AffineSpace(2, QQ)
sage: C = Curve(x^2+y-x)
sage: C = Curve(x^2 + y - x)
sage: enum_affine_rational_field(C, 10) # long time (3 s)
[(-2, -6), (-1, -2), (-2/3, -10/9), (-1/2, -3/4), (-1/3, -4/9),
(0, 0), (1/3, 2/9), (1/2, 1/4), (2/3, 2/9), (1, 0),
(4/3, -4/9), (3/2, -3/4), (5/3, -10/9), (2, -2), (3, -6)]
(0, 0), (1/3, 2/9), (1/2, 1/4), (2/3, 2/9), (1, 0),
(4/3, -4/9), (3/2, -3/4), (5/3, -10/9), (2, -2), (3, -6)]

AUTHORS:

Expand Down Expand Up @@ -183,31 +183,32 @@ def enum_affine_number_field(X, **kwds):

OUTPUT:

- a list containing the affine points of ``X`` of absolute height up to ``B``,
sorted.
- a list containing the affine points of ``X`` of absolute height up to ``B``,
sorted.

EXAMPLES::

sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 2, 'v')
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: X = A.subscheme([y^2 - x])
sage: enum_affine_number_field(X(K), bound=2**0.5)
[(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v), (0, 0, v), (0, 0, 1),
(1, -1, -1), (1, -1, -v), (1, -1, -1/2*v), (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1),
(1, 1, -1), (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]
sage: K = NumberField(u^2 + 2, 'v') # optional - sage.rings.number_field
sage: A.<x,y,z> = AffineSpace(K, 3) # optional - sage.rings.number_field
sage: X = A.subscheme([y^2 - x]) # optional - sage.rings.number_field
sage: enum_affine_number_field(X(K), bound=2**0.5) # optional - sage.rings.number_field
[(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v),
(0, 0, v), (0, 0, 1), (1, -1, -1), (1, -1, -v), (1, -1, -1/2*v),
(1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1), (1, 1, -1),
(1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]

::

sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = AffineSpace(K, 2)
sage: X=A.subscheme(x-y)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: enum_affine_number_field(X, bound=3**0.25)
[(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2), (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2),
(1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v') # optional - sage.rings.number_field
sage: A.<x,y> = AffineSpace(K, 2) # optional - sage.rings.number_field
sage: X = A.subscheme(x - y) # optional - sage.rings.number_field
sage: enum_affine_number_field(X, bound=3**0.25) # optional - sage.rings.number_field
[(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2),
(0, 0), (-1/2*v + 1/2, -1/2*v + 1/2), (1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]
"""
B = kwds.pop('bound')
tol = kwds.pop('tolerance', 1e-2)
Expand Down Expand Up @@ -248,31 +249,31 @@ def enum_affine_finite_field(X):

EXAMPLES::

sage: F = GF(7)
sage: A.<w,x,y,z> = AffineSpace(4, F)
sage: C = A.subscheme([w^2+x+4, y*z*x-6, z*y+w*x])
sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
sage: enum_affine_finite_field(C(F))
sage: F = GF(7) # optional - sage.rings.finite_rings
sage: A.<w,x,y,z> = AffineSpace(4, F) # optional - sage.rings.finite_rings
sage: C = A.subscheme([w^2 + x + 4, y*z*x - 6, z*y + w*x]) # optional - sage.rings.finite_rings
sage: enum_affine_finite_field(C(F)) # optional - sage.rings.finite_rings
[]
sage: C = A.subscheme([w^2+x+4, y*z*x-6])
sage: enum_affine_finite_field(C(F))
sage: C = A.subscheme([w^2 + x + 4, y*z*x - 6]) # optional - sage.rings.finite_rings
sage: enum_affine_finite_field(C(F)) # optional - sage.rings.finite_rings
[(0, 3, 1, 2), (0, 3, 2, 1), (0, 3, 3, 3), (0, 3, 4, 4), (0, 3, 5, 6),
(0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
(1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
(2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
(3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
(4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
(5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
(5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
(6, 2, 5, 2), (6, 2, 6, 4)]
(0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
(1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
(2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
(3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
(4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
(5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
(5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
(6, 2, 5, 2), (6, 2, 6, 4)]

::

sage: A.<x,y,z> = AffineSpace(3, GF(3))
sage: S = A.subscheme(x+y)
sage: enum_affine_finite_field(S)
sage: A.<x,y,z> = AffineSpace(3, GF(3)) # optional - sage.rings.finite_rings
sage: S = A.subscheme(x + y) # optional - sage.rings.finite_rings
sage: enum_affine_finite_field(S) # optional - sage.rings.finite_rings
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2),
(2, 1, 0), (2, 1, 1), (2, 1, 2)]
(2, 1, 0), (2, 1, 1), (2, 1, 2)]

ALGORITHM:

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