# Cython helper methods to compute integral points in polyhedra.¶

class sage.geometry.integral_points.InequalityCollection

Bases: object

A collection of inequalities.

INPUT:

• polyhedron – a polyhedron defining the inequalities.

• permutation – list; a 0-based permutation of the coordinates. Will be used to permute the coordinates of the inequality.

• box_min, box_max – the (not permuted) minimal and maximal coordinates of the bounding box. Used for bounds checking.

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection
sage: P_QQ = Polyhedron(identity_matrix(3).columns() + [(-2, -1,-1)], base_ring=QQ)
sage: ieq = InequalityCollection(P_QQ, [0,1,2], *3,*3); ieq
The collection of inequalities
integer: (3, -2, -2) x + 2 >= 0
integer: (-1, 4, -1) x + 1 >= 0
integer: (-1, -1, 4) x + 1 >= 0
integer: (-1, -1, -1) x + 1 >= 0

sage: P_RR = Polyhedron(identity_matrix(2).columns() + [(-2.7, -1)], base_ring=RDF)
sage: InequalityCollection(P_RR, [0,1], *2, *2)
The collection of inequalities
integer: (-1, -1) x + 1 >= 0
generic: (-1.0, 3.7) x + 1.0 >= 0
generic: (1.0, -1.35) x + 1.35 >= 0

sage: line = Polyhedron(eqns=[(2,3,7)])
sage: InequalityCollection(line, [0,1], *2, *2 )
The collection of inequalities
integer: (3, 7) x + 2 >= 0
integer: (-3, -7) x + -2 >= 0

are_satisfied(inner_loop_variable)

Return whether all inequalities are satisfied.

You must call prepare_inner_loop() before calling this method.

INPUT:

• inner_loop_variable – Integer. the 0-th coordinate of the lattice point.

OUTPUT:

Boolean. Whether the lattice point is in the polyhedron.

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection
sage: line = Polyhedron(eqns=[(2,3,7)])
sage: ieq = InequalityCollection(line, [0,1], *2, *2 )
sage: ieq.prepare_next_to_inner_loop([3,4])
sage: ieq.prepare_inner_loop([3,4])
sage: ieq.are_satisfied(3)
False

prepare_inner_loop(p)

Peel off the inner loop.

In the inner loop of rectangular_box_points(), we have to repeatedly evaluate $$A x+b\geq 0$$. To speed up computation, we pre-evaluate

$c = A x - A_0 x_0 +b = b + \sum_{i=1} A_i x_i$

and only test $$A_0 x_0 +c \geq 0$$ in the inner loop.

You must call prepare_next_to_inner_loop() before calling this method.

INPUT:

• p – the coordinates of the point to loop over. Only the p[1:] entries are used.

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection, print_cache
sage: P = Polyhedron(ieqs=[(2,3,7,11)])
sage: ieq = InequalityCollection(P, [0,1,2], *3,*3); ieq
The collection of inequalities
integer: (3, 7, 11) x + 2 >= 0
sage: ieq.prepare_next_to_inner_loop([2,1,3])
sage: ieq.prepare_inner_loop([2,1,3])
sage: print_cache(ieq)
Cached inner loop: 3 * x_0 + 42 >= 0
Cached next-to-inner loop: 3 * x_0 + 7 * x_1 + 35 >= 0

prepare_next_to_inner_loop(p)

Peel off the next-to-inner loop.

In the next-to-inner loop of rectangular_box_points(), we have to repeatedly evaluate $$A x-A_0 x_0+b$$. To speed up computation, we pre-evaluate

$c = b + \sum_{i=2} A_i x_i$

and only compute $$A x-A_0 x_0+b = A_1 x_1 +c \geq 0$$ in the next-to-inner loop.

INPUT:

• p – the point coordinates. Only p[2:] coordinates are potentially used by this method.

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection, print_cache
sage: P = Polyhedron(ieqs=[(2,3,7,11)])
sage: ieq = InequalityCollection(P, [0,1,2], *3,*3); ieq
The collection of inequalities
integer: (3, 7, 11) x + 2 >= 0
sage: ieq.prepare_next_to_inner_loop([2,1,3])
sage: ieq.prepare_inner_loop([2,1,3])
sage: print_cache(ieq)
Cached inner loop: 3 * x_0 + 42 >= 0
Cached next-to-inner loop: 3 * x_0 + 7 * x_1 + 35 >= 0

satisfied_as_equalities(inner_loop_variable)

Return the inequalities (by their index) that are satisfied as equalities.

INPUT:

• inner_loop_variable – Integer. the 0-th coordinate of the lattice point.

OUTPUT:

A set of integers in ascending order. Each integer is the index of a H-representation object of the polyhedron (either a inequality or an equation).

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection
sage: ieqs = InequalityCollection(quadrant, [0,1], [-1]*2, *2 )
sage: ieqs.prepare_next_to_inner_loop([-1,0])
sage: ieqs.prepare_inner_loop([-1,0])
sage: ieqs.satisfied_as_equalities(-1)
frozenset({1})
sage: ieqs.satisfied_as_equalities(0)
frozenset({0, 1})
sage: ieqs.satisfied_as_equalities(1)
frozenset({1})

swap_ineq_to_front(i)

Swap the i-th entry of the list to the front of the list of inequalities.

INPUT:

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection
sage: P_QQ = Polyhedron(identity_matrix(3).columns() + [(-2, -1,-1)], base_ring=QQ)
sage: iec = InequalityCollection(P_QQ, [0,1,2], *3,*3)
sage: iec
The collection of inequalities
integer: (3, -2, -2) x + 2 >= 0
integer: (-1, 4, -1) x + 1 >= 0
integer: (-1, -1, 4) x + 1 >= 0
integer: (-1, -1, -1) x + 1 >= 0
sage: iec.swap_ineq_to_front(3)
sage: iec
The collection of inequalities
integer: (-1, -1, -1) x + 1 >= 0
integer: (3, -2, -2) x + 2 >= 0
integer: (-1, 4, -1) x + 1 >= 0
integer: (-1, -1, 4) x + 1 >= 0

class sage.geometry.integral_points.Inequality_generic

Bases: object

An inequality whose coefficients are arbitrary Python/Sage objects

INPUT:

• A – list of coefficients

• b – element

OUTPUT:

Inequality $$A x + b \geq 0$$.

EXAMPLES:

sage: from sage.geometry.integral_points import Inequality_generic
sage: Inequality_generic([2*pi,sqrt(3),7/2], -5.5)
generic: (2*pi, sqrt(3), 7/2) x + -5.50000000000000 >= 0

class sage.geometry.integral_points.Inequality_int

Bases: object

Fast version of inequality in the case that all coefficients fit into machine ints.

INPUT:

• A – list of integers

• b – integer

• max_abs_coordinates – the maximum of the coordinates that one wants to evaluate the coordinates on; used for overflow checking

OUTPUT:

Inequality $$A x + b \geq 0$$. A OverflowError is raised if a machine integer is not long enough to hold the results. A ValueError is raised if some of the input is not integral.

EXAMPLES:

sage: from sage.geometry.integral_points import Inequality_int
sage: Inequality_int([2,3,7], -5, *3)
integer: (2, 3, 7) x + -5 >= 0

sage: Inequality_int(*21, -5, *21)
Traceback (most recent call last):
...
OverflowError: Dimension limit exceeded.

sage: Inequality_int([2,3/2,7], -5, *3)
Traceback (most recent call last):
...
ValueError: Not integral.

sage: Inequality_int([2,3,7], -5.2, *3)
Traceback (most recent call last):
...
ValueError: Not integral.

sage: Inequality_int([2,3,7], -5*10^50, *3)  # actual error message can differ between 32 and 64 bit
Traceback (most recent call last):
...
OverflowError: ...

sage.geometry.integral_points.loop_over_parallelotope_points(e, d, VDinv, R, lattice, A=None, b=None)

The inner loop of parallelotope_points().

INPUT:

See parallelotope_points() for e, d, VDinv, R, lattice.

• A, b: Either both None or a vector and number. If present, only the parallelotope points satisfying $$A x \leq b$$ are returned.

OUTPUT:

The points of the half-open parallelotope as a tuple of lattice points.

EXAMPLES:

sage: e = 
sage: d = prod(e)
sage: VDinv = matrix(ZZ, [])
sage: R = column_matrix(ZZ,[3,3,3])
sage: lattice = ZZ^3
sage: from sage.geometry.integral_points import loop_over_parallelotope_points
sage: loop_over_parallelotope_points(e, d, VDinv, R, lattice)
((0, 0, 0), (1, 1, 1), (2, 2, 2))

sage: A = vector(ZZ, [1,0,0])
sage: b = 1
sage: loop_over_parallelotope_points(e, d, VDinv, R, lattice, A, b)
((0, 0, 0), (1, 1, 1))

sage.geometry.integral_points.parallelotope_points(spanning_points, lattice)

Return integral points in the parallelotope starting at the origin and spanned by the spanning_points.

See semigroup_generators() for a description of the algorithm.

INPUT:

• spanning_points – a non-empty list of linearly independent rays ($$\ZZ$$-vectors or toric lattice elements), not necessarily primitive lattice points.

OUTPUT:

The tuple of all lattice points in the half-open parallelotope spanned by the rays $$r_i$$,

$\mathop{par}(\{r_i\}) = \sum_{0\leq a_i < 1} a_i r_i$

By half-open parallelotope, we mean that the points in the facets not meeting the origin are omitted.

EXAMPLES:

Note how the points on the outward-facing factes are omitted:

sage: from sage.geometry.integral_points import parallelotope_points
sage: rays = list(map(vector, [(2,0), (0,2)]))
sage: parallelotope_points(rays, ZZ^2)
((0, 0), (0, 1), (1, 0), (1, 1))


The rays can also be toric lattice points:

sage: rays = list(map(ToricLattice(2), [(2,0), (0,2)]))
sage: parallelotope_points(rays, ToricLattice(2))
(N(0, 0), N(0, 1), N(1, 0), N(1, 1))


A non-smooth cone:

sage: c = Cone([ (1,0), (1,2) ])
sage: parallelotope_points(c.rays(), c.lattice())
(N(0, 0), N(1, 1))


A ValueError is raised if the spanning_points are not linearly independent:

sage: rays = list(map(ToricLattice(2), [(1,1)]*2))
sage: parallelotope_points(rays, ToricLattice(2))
Traceback (most recent call last):
...
ValueError: The spanning points are not linearly independent!

sage.geometry.integral_points.print_cache(inequality_collection)

Print the cached values in Inequality_int (for debugging/doctesting only).

EXAMPLES:

sage: from sage.geometry.integral_points import InequalityCollection, print_cache
sage: P = Polyhedron(ieqs=[(2,3,7)])
sage: ieq = InequalityCollection(P, [0,1], *2,*2); ieq
The collection of inequalities
integer: (3, 7) x + 2 >= 0
sage: ieq.prepare_next_to_inner_loop([3,5])
sage: ieq.prepare_inner_loop([3,5])
sage: print_cache(ieq)
Cached inner loop: 3 * x_0 + 37 >= 0
Cached next-to-inner loop: 3 * x_0 + 7 * x_1 + 2 >= 0

sage.geometry.integral_points.ray_matrix_normal_form(R)

Compute the Smith normal form of the ray matrix for parallelotope_points().

INPUT:

• R$$\ZZ$$-matrix whose columns are the rays spanning the parallelotope.

OUTPUT:

A tuple containing e, d, and VDinv.

EXAMPLES:

sage: from sage.geometry.integral_points import ray_matrix_normal_form
sage: R = column_matrix(ZZ,[3,3,3])
sage: ray_matrix_normal_form(R)
(, 3, )

sage.geometry.integral_points.rectangular_box_points(box_min, box_max, polyhedron=None, count_only=False, return_saturated=False)

Return the integral points in the lattice bounding box that are also contained in the given polyhedron.

INPUT:

• box_min – A list of integers. The minimal value for each coordinate of the rectangular bounding box.

• box_max – A list of integers. The maximal value for each coordinate of the rectangular bounding box.

• polyhedron – A Polyhedron_base, a PPL C_Polyhedron, or None (default).

• count_only – Boolean (default: False). Whether to return only the total number of vertices, and not their coordinates. Enabling this option speeds up the enumeration. Cannot be combined with the return_saturated option.

• return_saturated – Boolean (default: False. Whether to also return which inequalities are saturated for each point of the polyhedron. Enabling this slows down the enumeration. Cannot be combined with the count_only option.

OUTPUT:

By default, this function returns a tuple containing the integral points of the rectangular box spanned by box_min and box_max and that lie inside the polyhedron. For sufficiently large bounding boxes, this are all integral points of the polyhedron.

If no polyhedron is specified, all integral points of the rectangular box are returned.

If count_only is specified, only the total number (an integer) of found lattice points is returned.

If return_saturated is enabled, then for each integral point a pair (point, Hrep) is returned where point is the point and Hrep is the set of indices of the H-representation objects that are saturated at the point.

ALGORITHM:

This function implements the naive algorithm towards counting integral points. Given min and max of vertex coordinates, it iterates over all points in the bounding box and checks whether they lie in the polyhedron. The following optimizations are implemented:

• Cython: Use machine integers and optimizing C/C++ compiler where possible, arbitrary precision integers where necessary. Bounds checking, no compile time limits.

• Unwind inner loop (and next-to-inner loop):

$Ax \leq b \quad \Leftrightarrow \quad a_1 x_1 ~\leq~ b - \sum_{i=2}^d a_i x_i$

so we only have to evaluate $$a_1 * x_1$$ in the inner loop.

• Coordinates are permuted to make the longest box edge the inner loop. The inner loop is optimized to run very fast, so its best to do as much work as possible there.

• Continuously reorder inequalities and test the most restrictive inequalities first.

• Use convexity and only find first and last allowed point in the inner loop. The points in-between must be points of the polyhedron, too.

EXAMPLES:

sage: from sage.geometry.integral_points import rectangular_box_points
sage: rectangular_box_points([0,0,0],[1,2,3])
((0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3),
(0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 1, 3),
(0, 2, 0), (0, 2, 1), (0, 2, 2), (0, 2, 3),
(1, 0, 0), (1, 0, 1), (1, 0, 2), (1, 0, 3),
(1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 1, 3),
(1, 2, 0), (1, 2, 1), (1, 2, 2), (1, 2, 3))

sage: from sage.geometry.integral_points import rectangular_box_points
sage: rectangular_box_points([0,0,0],[1,2,3], count_only=True)
24

sage: cell24 = polytopes.twenty_four_cell()
sage: rectangular_box_points([-1]*4, *4, cell24)
((-1, 0, 0, 0), (0, -1, 0, 0), (0, 0, -1, 0), (0, 0, 0, -1),
(0, 0, 0, 0),
(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0))
sage: d = 3
sage: dilated_cell24 = d*cell24
sage: len( rectangular_box_points([-d]*4, [d]*4, dilated_cell24) )
305

sage: d = 6
sage: dilated_cell24 = d*cell24
sage: len( rectangular_box_points([-d]*4, [d]*4, dilated_cell24) )
3625

sage: rectangular_box_points([-d]*4, [d]*4, dilated_cell24, count_only=True)
3625

sage: polytope = Polyhedron([(-4,-3,-2,-1),(3,1,1,1),(1,2,1,1),(1,1,3,0),(1,3,2,4)])
sage: pts = rectangular_box_points([-4]*4, *4, polytope); pts
((-4, -3, -2, -1), (-1, 0, 0, 1), (0, 1, 1, 1), (1, 1, 1, 1), (1, 1, 3, 0),
(1, 2, 1, 1), (1, 2, 2, 2), (1, 3, 2, 4), (2, 1, 1, 1), (3, 1, 1, 1))
sage: all(polytope.contains(p) for p in pts)
True

sage: set(map(tuple,pts)) == \
....: set([(-4,-3,-2,-1),(3,1,1,1),(1,2,1,1),(1,1,3,0),(1,3,2,4),
....:      (0,1,1,1),(1,2,2,2),(-1,0,0,1),(1,1,1,1),(2,1,1,1)])   # computed with PALP
True


Long ints and non-integral polyhedra are explicitly allowed:

sage: polytope = Polyhedron([, [10*pi.n()]], base_ring=RDF)
sage: len( rectangular_box_points([-100], , polytope) )
31

sage: halfplane = Polyhedron(ieqs=[(-1,1,0)])
sage: rectangular_box_points([0,-1+10^50], [0,1+10^50])
((0, 99999999999999999999999999999999999999999999999999),
(0, 100000000000000000000000000000000000000000000000000),
(0, 100000000000000000000000000000000000000000000000001))
sage: len( rectangular_box_points([0,-100+10^50], [1,100+10^50], halfplane) )
201


Using a PPL polyhedron:

sage: from ppl import Variable, Generator_System, C_Polyhedron, point
sage: gs = Generator_System()
sage: x = Variable(0); y = Variable(1); z = Variable(2)
sage: gs.insert(point(0*x + 1*y + 0*z))
sage: gs.insert(point(0*x + 1*y + 3*z))
sage: gs.insert(point(3*x + 1*y + 0*z))
sage: gs.insert(point(3*x + 1*y + 3*z))
sage: poly = C_Polyhedron(gs)
sage: rectangular_box_points(*3, *3, poly)
((0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 1, 3), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 1, 3),
(2, 1, 0), (2, 1, 1), (2, 1, 2), (2, 1, 3), (3, 1, 0), (3, 1, 1), (3, 1, 2), (3, 1, 3))


Optionally, return the information about the saturated inequalities as well:

sage: cube = polytopes.cube()
sage: cube.Hrepresentation(0)
An inequality (-1, 0, 0) x + 1 >= 0
sage: cube.Hrepresentation(1)
An inequality (0, -1, 0) x + 1 >= 0
sage: cube.Hrepresentation(2)
An inequality (0, 0, -1) x + 1 >= 0
sage: rectangular_box_points(*3, *3, cube, return_saturated=True)
(((0, 0, 0), frozenset()),
((0, 0, 1), frozenset({2})),
((0, 1, 0), frozenset({1})),
((0, 1, 1), frozenset({1, 2})),
((1, 0, 0), frozenset({0})),
((1, 0, 1), frozenset({0, 2})),
((1, 1, 0), frozenset({0, 1})),
((1, 1, 1), frozenset({0, 1, 2})))

sage.geometry.integral_points.simplex_points(vertices)

Return the integral points in a lattice simplex.

INPUT:

• vertices – an iterable of integer coordinate vectors. The indices of vertices that span the simplex under consideration.

OUTPUT:

A tuple containing the integral point coordinates as $$\ZZ$$-vectors.

EXAMPLES:

sage: from sage.geometry.integral_points import simplex_points
sage: simplex_points([(1,2,3), (2,3,7), (-2,-3,-11)])
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))


The simplex need not be full-dimensional:

sage: simplex = Polyhedron([(1,2,3,5), (2,3,7,5), (-2,-3,-11,5)])
sage: simplex_points(simplex.Vrepresentation())
((2, 3, 7, 5), (0, 0, -2, 5), (-2, -3, -11, 5), (1, 2, 3, 5))

sage: simplex_points([(2,3,7)])
((2, 3, 7),)