Zariski-Van Kampen method implementation#

This file contains functions to compute the fundamental group of the complement of a curve in the complex affine or projective plane, using Zariski-Van Kampen approach. It depends on the package sirocco.

The current implementation allows to compute a presentation of the fundamental group of curves over the rationals or number fields with a fixed embedding on \(\QQbar\).

Instead of computing a representation of the braid monodromy, we choose several base points and a system of paths joining them that generate all the necessary loops around the points of the discriminant. The group is generated by the free groups over these points, and braids over these paths give relations between these generators. This big group presentation is simplified at the end.

AUTHORS:

  • Miguel Marco (2015-09-30): Initial version

EXAMPLES:

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group, braid_monodromy
sage: R.<x, y> = QQ[]
sage: f = y^3 + x^3 - 1
sage: braid_monodromy(f)
([s1*s0, s1*s0, s1*s0], {0: 0, 1: 0, 2: 0}, {}, 3)
sage: fundamental_group(f)
Finitely presented group < x0 |  >

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group, braid_monodromy
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = y**Integer(3) + x**Integer(3) - Integer(1)
>>> braid_monodromy(f)
([s1*s0, s1*s0, s1*s0], {0: 0, 1: 0, 2: 0}, {}, 3)
>>> fundamental_group(f)
Finitely presented group < x0 |  >
sage.schemes.curves.zariski_vankampen.braid2rels(L)[source]#

Return a minimal set of relations of the group F / [(b * F([j])) / F([j]) for j in (1..d)] where F = FreeGroup(d) and b is a conjugate of a positive braid . One starts from the non-trivial relations determined by the positive braid and transform them in relations determined by b.

INPUT:

  • L – a tuple whose first element is a positive braid and the second element is a list of permutation braids.

OUTPUT:

A list of Tietze words for a minimal set of relations of F / [(g * b) / g for g in F.gens()].

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import braid2rels
sage: B.<s0, s1, s2> = BraidGroup(4)
sage: L = ((s1*s0)^2, [s2])
sage: braid2rels(L)
[(4, 1, -2, -1), (2, -4, -2, 1)]

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import braid2rels
>>> B = BraidGroup(Integer(4), names=('s0', 's1', 's2',)); (s0, s1, s2,) = B._first_ngens(3)
>>> L = ((s1*s0)**Integer(2), [s2])
>>> braid2rels(L)
[(4, 1, -2, -1), (2, -4, -2, 1)]
sage.schemes.curves.zariski_vankampen.braid_from_piecewise(strands)[source]#

Compute the braid corresponding to the piecewise linear curves strands.

INPUT:

  • strands – a list of lists of tuples (t, c1, c2), where t is a number between 0 and 1, and c1 and c2 are rationals or algebraic reals.

OUTPUT:

The braid formed by the piecewise linear strands.

EXAMPLES:

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import braid_from_piecewise
sage: paths = [[(0, 0, 1), (0.2, -1, -0.5), (0.8, -1, 0), (1, 0, -1)],
....:          [(0, -1, 0), (0.5, 0, -1), (1, 1, 0)],
....:          [(0, 1, 0), (0.5, 1, 1), (1, 0, 1)]]
sage: braid_from_piecewise(paths)
s0*s1

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import braid_from_piecewise
>>> paths = [[(Integer(0), Integer(0), Integer(1)), (RealNumber('0.2'), -Integer(1), -RealNumber('0.5')), (RealNumber('0.8'), -Integer(1), Integer(0)), (Integer(1), Integer(0), -Integer(1))],
...          [(Integer(0), -Integer(1), Integer(0)), (RealNumber('0.5'), Integer(0), -Integer(1)), (Integer(1), Integer(1), Integer(0))],
...          [(Integer(0), Integer(1), Integer(0)), (RealNumber('0.5'), Integer(1), Integer(1)), (Integer(1), Integer(0), Integer(1))]]
>>> braid_from_piecewise(paths)
s0*s1
sage.schemes.curves.zariski_vankampen.braid_monodromy(f, arrangement=(), vertical=False)[source]#

Compute the braid monodromy of a projection of the curve defined by a polynomial.

INPUT:

  • f – a polynomial with two variables, over a number field with an embedding in the complex numbers

  • arrangement – tuple (default: ()); an optional tuple of polynomials whose product equals f

  • vertical – boolean (default: False`); if set to ``True, arrangements contains more than one polynomial, some of them are of degree \(1\) in \(x\) and degree \(0\) in \(y\), and none of the other components have vertical asymptotes, then these components are marked as vertical and not used for the computation of the braid monodromy. The other ones are marked as horizontal. If a vertical component does not pass through a singular points of the projection of the horizontal components a trivial braid is added to the list.

OUTPUT:

  • A list of braids, images by the braid monodromy of a geometric basis of the complement of the discriminant of \(f\) in \(\CC\).

  • A dictionary: i, index of a strand is sent to the index of the corresponding factor in arrangement.

  • Another dictionary dv, only relevant if vertical is True. If j is the index of a braid corresponding to a vertical line with index i in arrangement, then dv[j] = i.

  • A non-negative integer: the number of strands of the braids, only necessary if the list of braids is empty.

Note

The projection over the \(x\) axis is used if there are no vertical asymptotes. Otherwise, a linear change of variables is done to fall into the previous case except if the only vertical asymptotes are lines and vertical=True.

EXAMPLES:

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import braid_monodromy
sage: R.<x, y> = QQ[]
sage: f = (x^2 - y^3) * (x + 3*y - 5)
sage: bm = braid_monodromy(f); bm
([s1*s0*(s1*s2)^2*s0*s2^2*s0^-1*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
  s1*s0*(s1*s2)^2*(s0*s2^-1*s1*s2*s1*s2^-1)^2*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
  s1*s0*(s1*s2)^2*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
  s1*s0*s2*s0^-1*s2*s1^-1], {0: 0, 1: 0, 2: 0, 3: 0}, {}, 4)
sage: flist = (x^2 - y^3, x + 3*y - 5)
sage: bm1 = braid_monodromy(f, arrangement=flist)
sage: bm1[0] == bm[0]
True
sage: bm1[1]
{0: 0, 1: 1, 2: 0, 3: 0}
sage: braid_monodromy(R(1))
([], {}, {}, 0)
sage: braid_monodromy(x*y^2 - 1)
([s0*s1*s0^-1*s1*s0*s1^-1*s0^-1, s0*s1*s0^-1, s0], {0: 0, 1: 0, 2: 0}, {}, 3)
sage: L = [x, y, x - 1, x -y]
sage: braid_monodromy(prod(L), arrangement=L, vertical=True)
([s^2, 1], {0: 1, 1: 3}, {0: 0, 1: 2}, 2)

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import braid_monodromy
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = (x**Integer(2) - y**Integer(3)) * (x + Integer(3)*y - Integer(5))
>>> bm = braid_monodromy(f); bm
([s1*s0*(s1*s2)^2*s0*s2^2*s0^-1*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
  s1*s0*(s1*s2)^2*(s0*s2^-1*s1*s2*s1*s2^-1)^2*(s2^-1*s1^-1)^2*s0^-1*s1^-1,
  s1*s0*(s1*s2)^2*s2*s1^-1*s2^-1*s1^-1*s0^-1*s1^-1,
  s1*s0*s2*s0^-1*s2*s1^-1], {0: 0, 1: 0, 2: 0, 3: 0}, {}, 4)
>>> flist = (x**Integer(2) - y**Integer(3), x + Integer(3)*y - Integer(5))
>>> bm1 = braid_monodromy(f, arrangement=flist)
>>> bm1[Integer(0)] == bm[Integer(0)]
True
>>> bm1[Integer(1)]
{0: 0, 1: 1, 2: 0, 3: 0}
>>> braid_monodromy(R(Integer(1)))
([], {}, {}, 0)
>>> braid_monodromy(x*y**Integer(2) - Integer(1))
([s0*s1*s0^-1*s1*s0*s1^-1*s0^-1, s0*s1*s0^-1, s0], {0: 0, 1: 0, 2: 0}, {}, 3)
>>> L = [x, y, x - Integer(1), x -y]
>>> braid_monodromy(prod(L), arrangement=L, vertical=True)
([s^2, 1], {0: 1, 1: 3}, {0: 0, 1: 2}, 2)
sage.schemes.curves.zariski_vankampen.conjugate_positive_form(braid)[source]#

For a braid which is conjugate to a product of disjoint positive braids a list of such decompositions is given.

INPUT:

  • braid – a braid \(\sigma\)

OUTPUT:

A list of \(r\) lists. Each such list is another list with two elements, a positive braid \(\alpha_i\) and a list of permutation braids \(\gamma_{1}^{i},\dots,\gamma_{r}^{n_i}\) such that if \(\gamma_i=\prod_{j=1}^{n_i} \gamma_j^i\) then the braids \(\tau_i=\gamma_i\alpha_i\gamma_i^{-1}\) pairwise commute and \(\alpha=\prod_{i=1}^{r} \tau_i\).

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import conjugate_positive_form
sage: B = BraidGroup(4)
sage: t = B((1, 3, 2, -3, 1, 1))
sage: conjugate_positive_form(t)
[[(s1*s0)^2, [s2]]]
sage: B = BraidGroup(5)
sage: t = B((1, 2, 3, 4, -1, -2, 3, 3, 2, -4))
sage: L = conjugate_positive_form(t); L
[[s1^2, [s3*s2]], [s1*s2, [s0]]]
sage: s = B.one()
sage: for a, l in L:
....:   b = prod(l)
....:   s *= b * a / b
sage: s == t
True
sage: s1 = B.gen(1)^3
sage: conjugate_positive_form(s1)
[[s1^3, []]]

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import conjugate_positive_form
>>> B = BraidGroup(Integer(4))
>>> t = B((Integer(1), Integer(3), Integer(2), -Integer(3), Integer(1), Integer(1)))
>>> conjugate_positive_form(t)
[[(s1*s0)^2, [s2]]]
>>> B = BraidGroup(Integer(5))
>>> t = B((Integer(1), Integer(2), Integer(3), Integer(4), -Integer(1), -Integer(2), Integer(3), Integer(3), Integer(2), -Integer(4)))
>>> L = conjugate_positive_form(t); L
[[s1^2, [s3*s2]], [s1*s2, [s0]]]
>>> s = B.one()
>>> for a, l in L:
...   b = prod(l)
...   s *= b * a / b
>>> s == t
True
>>> s1 = B.gen(Integer(1))**Integer(3)
>>> conjugate_positive_form(s1)
[[s1^3, []]]
sage.schemes.curves.zariski_vankampen.corrected_voronoi_diagram()[source]#

Compute a Voronoi diagram of a set of points with rational coordinates. The given points are granted to lie one in each bounded region.

INPUT:

  • points – a tuple of complex numbers

OUTPUT:

A Voronoi diagram constructed from rational approximations of the points, with the guarantee that each bounded region contains exactly one of the input points.

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import corrected_voronoi_diagram
sage: points = (2, I, 0.000001, 0, 0.000001*I)
sage: V = corrected_voronoi_diagram(points)
sage: V
The Voronoi diagram of 9 points of dimension 2 in the Rational Field
sage: V.regions()
{P(-7, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices and 2 rays,
P(0, -7): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices and 2 rays,
P(0, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices,
P(0, 1): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices,
P(0, 1/1000000): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices,
P(0, 7): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices and 2 rays,
P(1/1000000, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices,
P(2, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices,
P(7, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 2 rays}

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import corrected_voronoi_diagram
>>> points = (Integer(2), I, RealNumber('0.000001'), Integer(0), RealNumber('0.000001')*I)
>>> V = corrected_voronoi_diagram(points)
>>> V
The Voronoi diagram of 9 points of dimension 2 in the Rational Field
>>> V.regions()
{P(-7, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices and 2 rays,
P(0, -7): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices and 2 rays,
P(0, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices,
P(0, 1): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices,
P(0, 1/1000000): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices,
P(0, 7): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices and 2 rays,
P(1/1000000, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices,
P(2, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 5 vertices,
P(7, 0): A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 2 rays}
sage.schemes.curves.zariski_vankampen.discrim(pols)[source]#

Return the points in the discriminant of the product of the polynomials of a list or tuple pols.

The result is the set of values of the first variable for which two roots in the second variable coincide.

INPUT:

  • pols – a list or tuple of polynomials in two variables with coefficients in a number field with a fixed embedding in \(\QQbar\).

OUTPUT:

A tuple with the roots of the discriminant in \(\QQbar\).

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import discrim
sage: R.<x, y> = QQ[]
sage: flist = (y^3 + x^3 - 1, 2 * x + y)
sage: sorted((discrim(flist)))
[-0.522757958574711?,
 -0.500000000000000? - 0.866025403784439?*I,
 -0.500000000000000? + 0.866025403784439?*I,
 0.2613789792873551? - 0.4527216721561923?*I,
 0.2613789792873551? + 0.4527216721561923?*I,
 1]

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import discrim
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> flist = (y**Integer(3) + x**Integer(3) - Integer(1), Integer(2) * x + y)
>>> sorted((discrim(flist)))
[-0.522757958574711?,
 -0.500000000000000? - 0.866025403784439?*I,
 -0.500000000000000? + 0.866025403784439?*I,
 0.2613789792873551? - 0.4527216721561923?*I,
 0.2613789792873551? + 0.4527216721561923?*I,
 1]
sage.schemes.curves.zariski_vankampen.fieldI(field)[source]#

Return the (either double or trivial) extension of a number field which contains I.

INPUT:

  • field – a number field with an embedding in \(\QQbar\).

OUTPUT:

The extension F of field containing I with an embedding in \(\QQbar\).

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import fieldI
sage: p = QQ[x](x^5 + 2 * x + 1)
sage: a0 = p.roots(QQbar, multiplicities=False)[0]
sage: F0.<a> = NumberField(p, embedding=a0)
sage: fieldI(F0)
Number Field in prim with defining polynomial
x^10 + 5*x^8 + 14*x^6 - 2*x^5 - 10*x^4 + 20*x^3 - 11*x^2 - 14*x + 10
with prim = 0.4863890359345430? + 1.000000000000000?*I
sage: F0 = CyclotomicField(5)
sage: fieldI(F0)
Number Field in prim with defining polynomial
x^8 - 2*x^7 + 7*x^6 - 10*x^5 + 16*x^4 - 10*x^3 - 2*x^2 + 4*x + 1
with prim = -0.3090169943749474? + 0.04894348370484643?*I
sage: fieldI(QuadraticField(3))
Number Field in prim with defining polynomial x^4 - 4*x^2 + 16
with prim = -1.732050807568878? + 1.000000000000000?*I
sage: fieldI(QuadraticField(-3))
Number Field in prim with defining polynomial x^4 + 8*x^2 + 4
with prim = 0.?e-18 - 0.732050807568878?*I

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import fieldI
>>> p = QQ[x](x**Integer(5) + Integer(2) * x + Integer(1))
>>> a0 = p.roots(QQbar, multiplicities=False)[Integer(0)]
>>> F0 = NumberField(p, embedding=a0, names=('a',)); (a,) = F0._first_ngens(1)
>>> fieldI(F0)
Number Field in prim with defining polynomial
x^10 + 5*x^8 + 14*x^6 - 2*x^5 - 10*x^4 + 20*x^3 - 11*x^2 - 14*x + 10
with prim = 0.4863890359345430? + 1.000000000000000?*I
>>> F0 = CyclotomicField(Integer(5))
>>> fieldI(F0)
Number Field in prim with defining polynomial
x^8 - 2*x^7 + 7*x^6 - 10*x^5 + 16*x^4 - 10*x^3 - 2*x^2 + 4*x + 1
with prim = -0.3090169943749474? + 0.04894348370484643?*I
>>> fieldI(QuadraticField(Integer(3)))
Number Field in prim with defining polynomial x^4 - 4*x^2 + 16
with prim = -1.732050807568878? + 1.000000000000000?*I
>>> fieldI(QuadraticField(-Integer(3)))
Number Field in prim with defining polynomial x^4 + 8*x^2 + 4
with prim = 0.?e-18 - 0.732050807568878?*I

If I is already in the field, the result is the field itself:

sage: from sage.schemes.curves.zariski_vankampen import fieldI
sage: p = QQ[x](x^4 + 1)
sage: a0 = p.roots(QQbar, multiplicities=False)[0]
sage: F0.<a> = NumberField(p, embedding=a0)
sage: F1 = fieldI(F0)
sage: F0 == F1
True
sage: QuadraticField(-1) == fieldI(QuadraticField(-1))
True

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import fieldI
>>> p = QQ[x](x**Integer(4) + Integer(1))
>>> a0 = p.roots(QQbar, multiplicities=False)[Integer(0)]
>>> F0 = NumberField(p, embedding=a0, names=('a',)); (a,) = F0._first_ngens(1)
>>> F1 = fieldI(F0)
>>> F0 == F1
True
>>> QuadraticField(-Integer(1)) == fieldI(QuadraticField(-Integer(1)))
True
sage.schemes.curves.zariski_vankampen.followstrand(f, factors, x0, x1, y0a, prec=53)[source]#

Return a piecewise linear approximation of the homotopy continuation of the root y0a from x0 to x1.

INPUT:

  • f – an irreducible polynomial in two variables

  • factors – a list of irreducible polynomials in two variables

  • x0 – a complex value, where the homotopy starts

  • x1 – a complex value, where the homotopy ends

  • y0a – an approximate solution of the polynomial \(F(y) = f(x_0, y)\)

  • prec – the precision to use

OUTPUT:

A list of values \((t, y_{tr}, y_{ti})\) such that:

  • t is a real number between zero and one

  • \(f(t \cdot x_1 + (1-t) \cdot x_0, y_{tr} + I \cdot y_{ti})\) is zero (or a good enough approximation)

  • the piecewise linear path determined by the points has a tubular neighborhood where the actual homotopy continuation path lies, and no other root of f, nor any root of the polynomials in factors, intersects it.

EXAMPLES:

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import followstrand
sage: R.<x, y> = QQ[]
sage: f = x^2 + y^3
sage: x0 = CC(1, 0)
sage: x1 = CC(1, 0.5)
sage: followstrand(f, [], x0, x1, -1.0)             # abs tol 1e-15
[(0.0, -1.0, 0.0),
 (0.7500000000000001, -1.015090921153253, -0.24752813818386948),
 (1.0, -1.026166099551513, -0.32768940253604323)]
sage: fup = f.subs({y: y - 1/10})
sage: fdown = f.subs({y: y + 1/10})
sage: followstrand(f, [fup, fdown], x0, x1, -1.0)   # abs tol 1e-15
[(0.0, -1.0, 0.0),
 (0.5303300858899107, -1.0076747107983448, -0.17588022709184917),
 (0.7651655429449553, -1.015686131039112, -0.25243563967299404),
 (1.0, -1.026166099551513, -0.3276894025360433)]

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import followstrand
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(2) + y**Integer(3)
>>> x0 = CC(Integer(1), Integer(0))
>>> x1 = CC(Integer(1), RealNumber('0.5'))
>>> followstrand(f, [], x0, x1, -RealNumber('1.0'))             # abs tol 1e-15
[(0.0, -1.0, 0.0),
 (0.7500000000000001, -1.015090921153253, -0.24752813818386948),
 (1.0, -1.026166099551513, -0.32768940253604323)]
>>> fup = f.subs({y: y - Integer(1)/Integer(10)})
>>> fdown = f.subs({y: y + Integer(1)/Integer(10)})
>>> followstrand(f, [fup, fdown], x0, x1, -RealNumber('1.0'))   # abs tol 1e-15
[(0.0, -1.0, 0.0),
 (0.5303300858899107, -1.0076747107983448, -0.17588022709184917),
 (0.7651655429449553, -1.015686131039112, -0.25243563967299404),
 (1.0, -1.026166099551513, -0.3276894025360433)]
sage.schemes.curves.zariski_vankampen.fundamental_group(f, simplified=True, projective=False, puiseux=True)[source]#

Return a presentation of the fundamental group of the complement of the algebraic set defined by the polynomial f.

INPUT:

  • f – a polynomial in two variables, with coefficients in either the rationals or a number field with a fixed embedding in \(\QQbar\)

  • simplified – boolean (default: True); if set to True the presentation will be simplified (see below)

  • projective – boolean (default: False); if set to True, the fundamental group of the complement of the projective completion of the curve will be computed, otherwise, the fundamental group of the complement in the affine plane will be computed

  • puiseux – boolean (default: True); if set to True, a presentation of the fundamental group with the homotopy type of the complement of the affine curve is computed. If the Euler characteristic does not match, the homotopy type is obtained with a wedge of 2-spheres. One relation is added if projective is set to True.

If projective` is False and puiseux is True, a Zariski-VanKampen presentation is returned.

OUTPUT:

A presentation of the fundamental group of the complement of the curve defined by f.

EXAMPLES:

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group, braid_monodromy
sage: R.<x, y> = QQ[]
sage: f = x^2 + y^3
sage: fundamental_group(f)
Finitely presented group < x0, x1 | x0*x1^-1*x0^-1*x1^-1*x0*x1 >
sage: fundamental_group(f, simplified=False, puiseux=False).sorted_presentation()
Finitely presented group < x0, x1, x2 | x2^-1*x1^-1*x0*x1,
                                        x2^-1*x0*x1*x0^-1,
                                        x1^-1*x0^-1*x1^-1*x0*x1*x0 >
sage: fundamental_group(f, projective=True)
Finitely presented group < x0 | x0^3 >
sage: fundamental_group(f).sorted_presentation()
Finitely presented group < x0, x1 | x1^-1*x0^-1*x1^-1*x0*x1*x0 >

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group, braid_monodromy
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(2) + y**Integer(3)
>>> fundamental_group(f)
Finitely presented group < x0, x1 | x0*x1^-1*x0^-1*x1^-1*x0*x1 >
>>> fundamental_group(f, simplified=False, puiseux=False).sorted_presentation()
Finitely presented group < x0, x1, x2 | x2^-1*x1^-1*x0*x1,
                                        x2^-1*x0*x1*x0^-1,
                                        x1^-1*x0^-1*x1^-1*x0*x1*x0 >
>>> fundamental_group(f, projective=True)
Finitely presented group < x0 | x0^3 >
>>> fundamental_group(f).sorted_presentation()
Finitely presented group < x0, x1 | x1^-1*x0^-1*x1^-1*x0*x1*x0 >

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group
sage: R.<x, y> = QQ[]
sage: f = y^3 + x^3
sage: fundamental_group(f).sorted_presentation()
Finitely presented group < x0, x1, x2 | x2^-1*x1^-1*x0^-1*x2*x0*x1,
                                        x2^-1*x1^-1*x2*x0*x1*x0^-1 >

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = y**Integer(3) + x**Integer(3)
>>> fundamental_group(f).sorted_presentation()
Finitely presented group < x0, x1, x2 | x2^-1*x1^-1*x0^-1*x2*x0*x1,
                                        x2^-1*x1^-1*x2*x0*x1*x0^-1 >

It is also possible to have coefficients in a number field with a fixed embedding in \(\QQbar\):

sage: from sage.schemes.curves.zariski_vankampen import fundamental_group
sage: zeta = QQbar['x']('x^2 + x+ 1').roots(multiplicities=False)[0]
sage: zeta
-0.50000000000000000? - 0.866025403784439?*I
sage: F = NumberField(zeta.minpoly(), 'zeta', embedding=zeta)
sage: F.inject_variables()
Defining zeta
sage: R.<x, y> = F[]
sage: f = y^3 + x^3 + zeta * x + 1
sage: fundamental_group(f)                                  # needs sirocco
Finitely presented group < x0 |  >

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group
>>> zeta = QQbar['x']('x^2 + x+ 1').roots(multiplicities=False)[Integer(0)]
>>> zeta
-0.50000000000000000? - 0.866025403784439?*I
>>> F = NumberField(zeta.minpoly(), 'zeta', embedding=zeta)
>>> F.inject_variables()
Defining zeta
>>> R = F['x, y']; (x, y,) = R._first_ngens(2)
>>> f = y**Integer(3) + x**Integer(3) + zeta * x + Integer(1)
>>> fundamental_group(f)                                  # needs sirocco
Finitely presented group < x0 |  >

We compute the fundamental group of the complement of a quartic using the puiseux option:

sage: # optional - sirocco
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group
sage: R.<x, y> = QQ[]
sage: f = x^2 * y^2 + x^2 + y^2 - 2 * x * y  * (x + y + 1)
sage: g = fundamental_group(f); g.sorted_presentation()
Finitely presented group < x0, x1 | x1^-2*x0^2, (x1^-1*x0)^3 >
sage: g = fundamental_group(f, projective=True)
sage: g.order(), g.abelian_invariants()
(12, (4,))
sage: fundamental_group(y * (y - 1))
Finitely presented group < x0, x1 |  >

>>> from sage.all import *
>>> # optional - sirocco
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(2) * y**Integer(2) + x**Integer(2) + y**Integer(2) - Integer(2) * x * y  * (x + y + Integer(1))
>>> g = fundamental_group(f); g.sorted_presentation()
Finitely presented group < x0, x1 | x1^-2*x0^2, (x1^-1*x0)^3 >
>>> g = fundamental_group(f, projective=True)
>>> g.order(), g.abelian_invariants()
(12, (4,))
>>> fundamental_group(y * (y - Integer(1)))
Finitely presented group < x0, x1 |  >
sage.schemes.curves.zariski_vankampen.fundamental_group_arrangement(flist, simplified=True, projective=False, puiseux=True, vertical=False, braid_data=None)[source]#

Compute the fundamental group of the complement of a curve defined by a list of polynomials with the extra information about the correspondence of the generators and meridians of the elements of the list.

INPUT:

  • flist – a tuple of polynomial with two variables, over a number field with an embedding in the complex numbers

  • simplified – boolean (default: True); if set to True the presentation will be simplified (see below)

  • projective – boolean (default: False); if set to True, the fundamental group of the complement of the projective completion of the curve will be computed, otherwise, the fundamental group of the complement in the affine plane will be computed

  • puiseux – boolean (default: True); if set to True a presentation of the fundamental group with the homotopy type of the complement of the affine curve will be computed, adding one relation if projective is set to True.

  • vertical – boolean (default: False); if set to True, whenever no curve has vertical asymptotes the computation of braid monodromy is simpler if some lines are vertical

  • braid_data – tuple (default: None); if it is not the default it is the output of fundamental_group_from_braid_mon previously computed

OUTPUT:

  • A list of braids. The braids correspond to paths based in the same point; each of this paths is the conjugated of a loop around one of the points in the discriminant of the projection of f.

  • A dictionary attaching to j a tuple a list of elements of the group which are meridians of the curve in position j. If projective is False and the \(y\)-degree of the horizontal components coincide with the total degree, another key is added to give a meridian of the line at infinity.

EXAMPLES:

sage: # needs sirocco
sage: from sage.schemes.curves.zariski_vankampen import braid_monodromy
sage: from sage.schemes.curves.zariski_vankampen import fundamental_group_arrangement
sage: R.<x, y> = QQ[]
sage: flist = [x^2 - y^3, x + 3 * y - 5]
sage: g, dic = fundamental_group_arrangement(flist)
sage: g.sorted_presentation()
Finitely presented group
 < x0, x1, x2 | x2^-1*x1^-1*x2*x1, x2^-1*x0^-1*x2^-1*x0*x2*x0,
                x1^-1*x0^-1*x1*x0 >
sage: dic
{0: [x0, x2], 1: [x1], 2: [x0^-1*x2^-1*x1^-1*x0^-1]}
sage: g, dic = fundamental_group_arrangement(flist, simplified=False, puiseux=False)
sage: g.sorted_presentation(), dic
(Finitely presented group
 < x0, x1, x2, x3 | 1, 1, 1, 1, 1, 1, 1,
                    x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                    x3^-1*x2^-1*x1^-1*x0^-1*x1*x2*x3*x2,
                    x3^-1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-1*x0*x1*x2,
                    x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                    x3^-1*x1^-1*x0*x1,
                    x1^-1*x0^-1*x1*x0, x1^-1*x0^-1*x1*x0,
                    x1^-1*x0^-1*x1*x0, x1^-1*x0^-1*x1*x0 >,
 {0: [x0, x2, x3], 1: [x1], 2: [x3^-1*x2^-1*x1^-1*x0^-1]})
sage: fundamental_group_arrangement(flist, projective=True)
(Finitely presented group < x |  >, {0: [x], 1: [x^-3]})
sage: fundamental_group_arrangement([])
(Finitely presented group <  |  >, {})
sage: g, dic = fundamental_group_arrangement([x * y])
sage: g.sorted_presentation(), dic
(Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >,
 {0: [x0, x1], 1: [x1^-1*x0^-1]})
sage: fundamental_group_arrangement([y + x^2])
(Finitely presented group < x |  >, {0: [x]})
sage: fundamental_group_arrangement([y^2 + x], projective=True)
(Finitely presented group < x | x^2 >, {0: [x]})
sage: L = [x, y, x - 1, x -y]
sage: G, dic =fundamental_group_arrangement(L)
sage: G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x3*x2, x3^-1*x1^-1*x0^-1*x1*x3*x0,
                   x3^-1*x1^-1*x3*x0*x1*x0^-1, x2^-1*x0^-1*x2*x0 >
sage: dic
{0: [x1], 1: [x3], 2: [x2], 3: [x0], 4: [x3^-1*x2^-1*x1^-1*x0^-1]}
sage: fundamental_group_arrangement(L, vertical=True)
(Finitely presented group
 < x0, x1, x2, x3 | x3*x0*x3^-1*x0^-1, x3*x1*x3^-1*x1^-1,
                    x1*x2*x0*x2^-1*x1^-1*x0^-1,
                    x1*x2*x0*x1^-1*x0^-1*x2^-1 >,
 {0: [x2], 1: [x0], 2: [x3], 3: [x1], 4: [x3^-1*x2^-1*x1^-1*x0^-1]})

>>> from sage.all import *
>>> # needs sirocco
>>> from sage.schemes.curves.zariski_vankampen import braid_monodromy
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group_arrangement
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> flist = [x**Integer(2) - y**Integer(3), x + Integer(3) * y - Integer(5)]
>>> g, dic = fundamental_group_arrangement(flist)
>>> g.sorted_presentation()
Finitely presented group
 < x0, x1, x2 | x2^-1*x1^-1*x2*x1, x2^-1*x0^-1*x2^-1*x0*x2*x0,
                x1^-1*x0^-1*x1*x0 >
>>> dic
{0: [x0, x2], 1: [x1], 2: [x0^-1*x2^-1*x1^-1*x0^-1]}
>>> g, dic = fundamental_group_arrangement(flist, simplified=False, puiseux=False)
>>> g.sorted_presentation(), dic
(Finitely presented group
 < x0, x1, x2, x3 | 1, 1, 1, 1, 1, 1, 1,
                    x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                    x3^-1*x2^-1*x1^-1*x0^-1*x1*x2*x3*x2,
                    x3^-1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-1*x0*x1*x2,
                    x3^-1*x2^-1*x1^-1*x2*x3*x2^-1*x1*x2,
                    x3^-1*x1^-1*x0*x1,
                    x1^-1*x0^-1*x1*x0, x1^-1*x0^-1*x1*x0,
                    x1^-1*x0^-1*x1*x0, x1^-1*x0^-1*x1*x0 >,
 {0: [x0, x2, x3], 1: [x1], 2: [x3^-1*x2^-1*x1^-1*x0^-1]})
>>> fundamental_group_arrangement(flist, projective=True)
(Finitely presented group < x |  >, {0: [x], 1: [x^-3]})
>>> fundamental_group_arrangement([])
(Finitely presented group <  |  >, {})
>>> g, dic = fundamental_group_arrangement([x * y])
>>> g.sorted_presentation(), dic
(Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >,
 {0: [x0, x1], 1: [x1^-1*x0^-1]})
>>> fundamental_group_arrangement([y + x**Integer(2)])
(Finitely presented group < x |  >, {0: [x]})
>>> fundamental_group_arrangement([y**Integer(2) + x], projective=True)
(Finitely presented group < x | x^2 >, {0: [x]})
>>> L = [x, y, x - Integer(1), x -y]
>>> G, dic =fundamental_group_arrangement(L)
>>> G.sorted_presentation()
Finitely presented group
< x0, x1, x2, x3 | x3^-1*x2^-1*x3*x2, x3^-1*x1^-1*x0^-1*x1*x3*x0,
                   x3^-1*x1^-1*x3*x0*x1*x0^-1, x2^-1*x0^-1*x2*x0 >
>>> dic
{0: [x1], 1: [x3], 2: [x2], 3: [x0], 4: [x3^-1*x2^-1*x1^-1*x0^-1]}
>>> fundamental_group_arrangement(L, vertical=True)
(Finitely presented group
 < x0, x1, x2, x3 | x3*x0*x3^-1*x0^-1, x3*x1*x3^-1*x1^-1,
                    x1*x2*x0*x2^-1*x1^-1*x0^-1,
                    x1*x2*x0*x1^-1*x0^-1*x2^-1 >,
 {0: [x2], 1: [x0], 2: [x3], 3: [x1], 4: [x3^-1*x2^-1*x1^-1*x0^-1]})
sage.schemes.curves.zariski_vankampen.fundamental_group_from_braid_mon(bm, degree=None, simplified=True, projective=False, puiseux=True, vertical=[])[source]#

Return a presentation of the fundamental group computed from a braid monodromy.

INPUT:

  • bm – a list of braids

  • degree – integer (default: None); only needed if the braid monodromy is an empty list.

  • simplified – boolean (default: True); if set to True the presentation will be simplified (see below)

  • projective – boolean (default: False); if set to True, the fundamental group of the complement of the projective completion of the curve will be computed, otherwise, the fundamental group of the complement in the affine plane will be computed

  • puiseux – boolean (default: True); if set to True a presentation of the fundamental group with the homotopy type of the complement of the affine curve will be computed, adding one relation if projective is set to True.

  • vertical – list of integers (default: []); the indices in [0 .. r - 1] of the braids that surround a vertical line

If projective` is False and puiseux is True, a Zariski-VanKampen presentation is returned.

OUTPUT:

A presentation of the fundamental group of the complement of the union of the curve with some vertical lines from its braid monodromy.

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import fundamental_group_from_braid_mon
sage: B.<s0, s1, s2> = BraidGroup(4)
sage: bm = [s1*s2*s0*s1*s0^-1*s1^-1*s0^-1,
....:       s0*s1^2*s0*s2*s1*(s0^-1*s1^-1)^2*s0^-1,
....:       (s0*s1)^2]
sage: g = fundamental_group_from_braid_mon(bm, projective=True); g      # needs sirocco
Finitely presented group
< x1, x3 | x3^2*x1^2, x1^-1*x3^-1*x1*x3^-1*x1^-1*x3^-1 >
sage: print(g.order(), g.abelian_invariants())                         # needs sirocco
12 (4,)
sage: B2 = BraidGroup(2)
sage: bm = [B2(3 * [1])]
sage: g = fundamental_group_from_braid_mon(bm, vertical=[0]); g         # needs sirocco
Finitely presented group
< x0, x1, x2 | x2*x0*x1*x2^-1*x1^-1*x0^-1,
               x2*x0*x1*x0*x1^-1*x0^-1*x2^-1*x1^-1 >
sage: fundamental_group_from_braid_mon([]) is None                      # needs sirocco
True
sage: fundamental_group_from_braid_mon([], degree=2)                    # needs sirocco
Finitely presented group < x0, x1 |  >
sage: fundamental_group_from_braid_mon([SymmetricGroup(1).one()])       # needs sirocco
Finitely presented group < x |  >

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import fundamental_group_from_braid_mon
>>> B = BraidGroup(Integer(4), names=('s0', 's1', 's2',)); (s0, s1, s2,) = B._first_ngens(3)
>>> bm = [s1*s2*s0*s1*s0**-Integer(1)*s1**-Integer(1)*s0**-Integer(1),
...       s0*s1**Integer(2)*s0*s2*s1*(s0**-Integer(1)*s1**-Integer(1))**Integer(2)*s0**-Integer(1),
...       (s0*s1)**Integer(2)]
>>> g = fundamental_group_from_braid_mon(bm, projective=True); g      # needs sirocco
Finitely presented group
< x1, x3 | x3^2*x1^2, x1^-1*x3^-1*x1*x3^-1*x1^-1*x3^-1 >
>>> print(g.order(), g.abelian_invariants())                         # needs sirocco
12 (4,)
>>> B2 = BraidGroup(Integer(2))
>>> bm = [B2(Integer(3) * [Integer(1)])]
>>> g = fundamental_group_from_braid_mon(bm, vertical=[Integer(0)]); g         # needs sirocco
Finitely presented group
< x0, x1, x2 | x2*x0*x1*x2^-1*x1^-1*x0^-1,
               x2*x0*x1*x0*x1^-1*x0^-1*x2^-1*x1^-1 >
>>> fundamental_group_from_braid_mon([]) is None                      # needs sirocco
True
>>> fundamental_group_from_braid_mon([], degree=Integer(2))                    # needs sirocco
Finitely presented group < x0, x1 |  >
>>> fundamental_group_from_braid_mon([SymmetricGroup(Integer(1)).one()])       # needs sirocco
Finitely presented group < x |  >
sage.schemes.curves.zariski_vankampen.geometric_basis(G, E, EC0, p, dual_graph, vertical_regions={})[source]#

Return a geometric basis, based on a vertex.

INPUT:

  • G – a graph with the bounded edges of a Voronoi Diagram

  • E – a subgraph of G which is a cycle containing the bounded edges touching an unbounded region of a Voronoi Diagram

  • EC0 – A counterclockwise orientation of the vertices of E

  • p – a vertex of E

  • dual_graph – a dual graph for a plane embedding of G such that E is the boundary of the non-bounded component of the complement. The edges are labelled as the dual edges and the vertices are labelled by a tuple whose first element is the an integer for the position and the second one is the cyclic ordered list of vertices in the region

  • vertical_regions – dictionary (default: \({}\)); its keys are the vertices of dual_graph to fix regions associated with vertical lines

OUTPUT: A geometric basis and a dictionary.

The geometric basis is formed by a list of sequences of paths. Each path is a ist of vertices, that form a closed path in G, based at p, that goes to a region, surrounds it, and comes back by the same path it came. The concatenation of all these paths is equivalent to E.

The dictionary associates to each vertical line the index of the generator of the geometric basis associated to it.

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import geometric_basis, corrected_voronoi_diagram, voronoi_cells
sage: points = (0, -1, I, 1, -I)
sage: V = corrected_voronoi_diagram(points)
sage: G, E, p, EC, DG, VR = voronoi_cells(V, vertical_lines=frozenset((0 .. 4)))
sage: gb, vd = geometric_basis(G, E, EC, p, DG, vertical_regions=VR)
sage: gb
[[A vertex at (5/2, -5/2), A vertex at (5/2, 5/2), A vertex at (-5/2, 5/2),
  A vertex at (-1/2, 1/2), A vertex at (-1/2, -1/2), A vertex at (1/2, -1/2),
  A vertex at (1/2, 1/2), A vertex at (-1/2, 1/2), A vertex at (-5/2, 5/2),
  A vertex at (5/2, 5/2), A vertex at (5/2, -5/2)],
 [A vertex at (5/2, -5/2), A vertex at (5/2, 5/2), A vertex at (-5/2, 5/2),
  A vertex at (-1/2, 1/2), A vertex at (1/2, 1/2), A vertex at (5/2, 5/2),
  A vertex at (5/2, -5/2)],
 [A vertex at (5/2, -5/2), A vertex at (5/2, 5/2), A vertex at (1/2, 1/2),
  A vertex at (1/2, -1/2), A vertex at (5/2, -5/2)], [A vertex at (5/2, -5/2),
  A vertex at (1/2, -1/2), A vertex at (-1/2, -1/2), A vertex at (-1/2, 1/2),
  A vertex at (-5/2, 5/2), A vertex at (-5/2, -5/2), A vertex at (-1/2, -1/2),
  A vertex at (1/2, -1/2), A vertex at (5/2, -5/2)],
 [A vertex at (5/2, -5/2), A vertex at (1/2, -1/2), A vertex at (-1/2, -1/2),
  A vertex at (-5/2, -5/2), A vertex at (5/2, -5/2)]]
sage: vd
{0: 0, 1: 3, 2: 1, 3: 2, 4: 4}

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import geometric_basis, corrected_voronoi_diagram, voronoi_cells
>>> points = (Integer(0), -Integer(1), I, Integer(1), -I)
>>> V = corrected_voronoi_diagram(points)
>>> G, E, p, EC, DG, VR = voronoi_cells(V, vertical_lines=frozenset((ellipsis_iter(Integer(0) ,Ellipsis, Integer(4)))))
>>> gb, vd = geometric_basis(G, E, EC, p, DG, vertical_regions=VR)
>>> gb
[[A vertex at (5/2, -5/2), A vertex at (5/2, 5/2), A vertex at (-5/2, 5/2),
  A vertex at (-1/2, 1/2), A vertex at (-1/2, -1/2), A vertex at (1/2, -1/2),
  A vertex at (1/2, 1/2), A vertex at (-1/2, 1/2), A vertex at (-5/2, 5/2),
  A vertex at (5/2, 5/2), A vertex at (5/2, -5/2)],
 [A vertex at (5/2, -5/2), A vertex at (5/2, 5/2), A vertex at (-5/2, 5/2),
  A vertex at (-1/2, 1/2), A vertex at (1/2, 1/2), A vertex at (5/2, 5/2),
  A vertex at (5/2, -5/2)],
 [A vertex at (5/2, -5/2), A vertex at (5/2, 5/2), A vertex at (1/2, 1/2),
  A vertex at (1/2, -1/2), A vertex at (5/2, -5/2)], [A vertex at (5/2, -5/2),
  A vertex at (1/2, -1/2), A vertex at (-1/2, -1/2), A vertex at (-1/2, 1/2),
  A vertex at (-5/2, 5/2), A vertex at (-5/2, -5/2), A vertex at (-1/2, -1/2),
  A vertex at (1/2, -1/2), A vertex at (5/2, -5/2)],
 [A vertex at (5/2, -5/2), A vertex at (1/2, -1/2), A vertex at (-1/2, -1/2),
  A vertex at (-5/2, -5/2), A vertex at (5/2, -5/2)]]
>>> vd
{0: 0, 1: 3, 2: 1, 3: 2, 4: 4}
sage.schemes.curves.zariski_vankampen.newton(f, x0, i0)[source]#

Return the interval Newton operator.

INPUT:

  • f – a univariate polynomial

  • x0 – a number

  • I0 – an interval

OUTPUT:

The interval \(x_0-\frac{f(x_0)}{f'(I_0)}\)

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import newton
sage: R.<x> = QQbar[]
sage: f = x^3 + x
sage: x0 = 1/10
sage: I0 = RIF((-1/5,1/5))
sage: n = newton(f, x0, I0)
sage: n
0.0?
sage: n.real().endpoints()
(-0.0460743801652894, 0.0291454081632654)
sage: n.imag().endpoints()
(0.000000000000000, -0.000000000000000)

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import newton
>>> R = QQbar['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(3) + x
>>> x0 = Integer(1)/Integer(10)
>>> I0 = RIF((-Integer(1)/Integer(5),Integer(1)/Integer(5)))
>>> n = newton(f, x0, I0)
>>> n
0.0?
>>> n.real().endpoints()
(-0.0460743801652894, 0.0291454081632654)
>>> n.imag().endpoints()
(0.000000000000000, -0.000000000000000)
sage.schemes.curves.zariski_vankampen.orient_circuit(circuit, convex=False, precision=53, verbose=False)[source]#

Reverse a circuit if it goes clockwise; otherwise leave it unchanged.

INPUT:

  • circuit – a circuit in the graph of a Voronoi Diagram, given by a list of edges

  • convex – boolean (default: False); if set to True a simpler computation is made

  • precision – bits of precision (default: 53)

  • verbose – boolean (default: False); for testing purposes

OUTPUT:

The same circuit if it goes counterclockwise, and its reversed otherwise, given as the ordered list of vertices with identic extremities.

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import orient_circuit
sage: points = [(-4, 0), (4, 0), (0, 4), (0, -4), (0, 0)]
sage: V = VoronoiDiagram(points)
sage: E = Graph()
sage: for reg  in V.regions().values():
....:     if reg.rays() or reg.lines():
....:         E  = E.union(reg.vertex_graph())
sage: E.vertices(sort=True)
[A vertex at (-2, -2),
 A vertex at (-2, 2),
 A vertex at (2, -2),
 A vertex at (2, 2)]
sage: cir = E.eulerian_circuit()
sage: cir
[(A vertex at (-2, -2), A vertex at (2, -2), None),
 (A vertex at (2, -2), A vertex at (2, 2), None),
 (A vertex at (2, 2), A vertex at (-2, 2), None),
 (A vertex at (-2, 2), A vertex at (-2, -2), None)]
sage: cir_oriented = orient_circuit(cir); cir_oriented
(A vertex at (-2, -2), A vertex at (2, -2), A vertex at (2, 2),
 A vertex at (-2, 2), A vertex at (-2, -2))
sage: cirinv = list(reversed([(c[1],c[0],c[2]) for c in cir]))
sage: cirinv
[(A vertex at (-2, -2), A vertex at (-2, 2), None),
 (A vertex at (-2, 2), A vertex at (2, 2), None),
 (A vertex at (2, 2), A vertex at (2, -2), None),
 (A vertex at (2, -2), A vertex at (-2, -2), None)]
sage: orient_circuit(cirinv) == cir_oriented
True
sage: cir_oriented == orient_circuit(cir, convex=True)
True
sage: P0=[(1,1/2),(0,1),(1,1)]; P1=[(0,3/2),(-1,0)]
sage: Q=Polyhedron(P0).vertices()
sage: Q = [Q[2], Q[0], Q[1]] + [_ for _ in reversed(Polyhedron(P1).vertices())]
sage: Q
[A vertex at (1, 1/2), A vertex at (0, 1), A vertex at (1, 1),
 A vertex at (0, 3/2), A vertex at (-1, 0)]
sage: E = Graph()
sage: for v, w in zip(Q, Q[1:] + [Q[0]]):
....:   E.add_edge((v, w))
sage: cir = orient_circuit(E.eulerian_circuit(), precision=1, verbose=True)
2
sage: cir
(A vertex at (1, 1/2), A vertex at (0, 1), A vertex at (1, 1),
 A vertex at (0, 3/2), A vertex at (-1, 0), A vertex at (1, 1/2))

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import orient_circuit
>>> points = [(-Integer(4), Integer(0)), (Integer(4), Integer(0)), (Integer(0), Integer(4)), (Integer(0), -Integer(4)), (Integer(0), Integer(0))]
>>> V = VoronoiDiagram(points)
>>> E = Graph()
>>> for reg  in V.regions().values():
...     if reg.rays() or reg.lines():
...         E  = E.union(reg.vertex_graph())
>>> E.vertices(sort=True)
[A vertex at (-2, -2),
 A vertex at (-2, 2),
 A vertex at (2, -2),
 A vertex at (2, 2)]
>>> cir = E.eulerian_circuit()
>>> cir
[(A vertex at (-2, -2), A vertex at (2, -2), None),
 (A vertex at (2, -2), A vertex at (2, 2), None),
 (A vertex at (2, 2), A vertex at (-2, 2), None),
 (A vertex at (-2, 2), A vertex at (-2, -2), None)]
>>> cir_oriented = orient_circuit(cir); cir_oriented
(A vertex at (-2, -2), A vertex at (2, -2), A vertex at (2, 2),
 A vertex at (-2, 2), A vertex at (-2, -2))
>>> cirinv = list(reversed([(c[Integer(1)],c[Integer(0)],c[Integer(2)]) for c in cir]))
>>> cirinv
[(A vertex at (-2, -2), A vertex at (-2, 2), None),
 (A vertex at (-2, 2), A vertex at (2, 2), None),
 (A vertex at (2, 2), A vertex at (2, -2), None),
 (A vertex at (2, -2), A vertex at (-2, -2), None)]
>>> orient_circuit(cirinv) == cir_oriented
True
>>> cir_oriented == orient_circuit(cir, convex=True)
True
>>> P0=[(Integer(1),Integer(1)/Integer(2)),(Integer(0),Integer(1)),(Integer(1),Integer(1))]; P1=[(Integer(0),Integer(3)/Integer(2)),(-Integer(1),Integer(0))]
>>> Q=Polyhedron(P0).vertices()
>>> Q = [Q[Integer(2)], Q[Integer(0)], Q[Integer(1)]] + [_ for _ in reversed(Polyhedron(P1).vertices())]
>>> Q
[A vertex at (1, 1/2), A vertex at (0, 1), A vertex at (1, 1),
 A vertex at (0, 3/2), A vertex at (-1, 0)]
>>> E = Graph()
>>> for v, w in zip(Q, Q[Integer(1):] + [Q[Integer(0)]]):
...   E.add_edge((v, w))
>>> cir = orient_circuit(E.eulerian_circuit(), precision=Integer(1), verbose=True)
2
>>> cir
(A vertex at (1, 1/2), A vertex at (0, 1), A vertex at (1, 1),
 A vertex at (0, 3/2), A vertex at (-1, 0), A vertex at (1, 1/2))
sage.schemes.curves.zariski_vankampen.populate_roots_interval_cache(inputs)[source]#

Call roots_interval() to the inputs that have not been computed previously, and cache them.

INPUT:

  • inputs – a list of tuples (f, x0)

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import populate_roots_interval_cache, roots_interval_cache, fieldI
sage: R.<x,y> = QQ[]
sage: K=fieldI(QQ)
sage: f = y^5 - x^2
sage: f = f.change_ring(K)
sage: (f, 3) in roots_interval_cache
False
sage: populate_roots_interval_cache([(f, 3)])
sage: (f, 3) in roots_interval_cache
True
sage: roots_interval_cache[(f, 3)]
{-1.255469441943070? - 0.9121519421827974?*I: -2.? - 1.?*I,
 -1.255469441943070? + 0.9121519421827974?*I: -2.? + 1.?*I,
 0.4795466549853897? - 1.475892845355996?*I: 1.? - 2.?*I,
 0.4795466549853897? + 1.475892845355996?*I: 1.? + 2.?*I,
 14421467174121563/9293107134194871: 2.? + 0.?*I}

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import populate_roots_interval_cache, roots_interval_cache, fieldI
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> K=fieldI(QQ)
>>> f = y**Integer(5) - x**Integer(2)
>>> f = f.change_ring(K)
>>> (f, Integer(3)) in roots_interval_cache
False
>>> populate_roots_interval_cache([(f, Integer(3))])
>>> (f, Integer(3)) in roots_interval_cache
True
>>> roots_interval_cache[(f, Integer(3))]
{-1.255469441943070? - 0.9121519421827974?*I: -2.? - 1.?*I,
 -1.255469441943070? + 0.9121519421827974?*I: -2.? + 1.?*I,
 0.4795466549853897? - 1.475892845355996?*I: 1.? - 2.?*I,
 0.4795466549853897? + 1.475892845355996?*I: 1.? + 2.?*I,
 14421467174121563/9293107134194871: 2.? + 0.?*I}
sage.schemes.curves.zariski_vankampen.roots_interval_cached(f, x0)[source]#

Cached version of roots_interval().

sage.schemes.curves.zariski_vankampen.strand_components(f, pols, p1)[source]#

Compute only the assignment from strands to elements of flist.

INPUT:

  • f – a reduced polynomial with two variables, over a number field with an embedding in the complex numbers

  • pols – a list of polynomials with two variables whose product equals f

  • p1 – a Gauss rational

OUTPUT:

  • A list and a dictionary. The first one is an ordered list of pairs consisting of (z,i) where z is a root of f(p_1,y) and \(i\) is the position of the polynomial in the list whose root is z. The second one attaches a number \(i\) (strand) to a number \(j\) (a polynomial in the list).

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import strand_components
sage: R.<x, y> = QQ[]
sage: flist = [x^2 - y^3, x + 3 * y - 5]
sage: strand_components(prod(flist), flist, 1)
([(-0.500000000000000? - 0.866025403784439?*I, 0),
  (-0.500000000000000? + 0.866025403784439?*I, 0),
  (1, 0), (1.333333333333334?, 1)], {0: 0, 1: 0, 2: 0, 3: 1})

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import strand_components
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> flist = [x**Integer(2) - y**Integer(3), x + Integer(3) * y - Integer(5)]
>>> strand_components(prod(flist), flist, Integer(1))
([(-0.500000000000000? - 0.866025403784439?*I, 0),
  (-0.500000000000000? + 0.866025403784439?*I, 0),
  (1, 0), (1.333333333333334?, 1)], {0: 0, 1: 0, 2: 0, 3: 1})
sage.schemes.curves.zariski_vankampen.vertical_lines_in_braidmon(pols)[source]#

Return the vertical lines in pols, unless one of the other components has a vertical asymptote.

INPUT:

  • pols – a list of polynomials with two variables whose product equals f

OUTPUT:

A list with the indices of the vertical lines in flist if there is no other componnet with vertical asymptote; otherwise it returns an empty list.

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import vertical_lines_in_braidmon
sage: R.<x, y> = QQ[]
sage: flist = [x^2 - y^3, x, x + 3 * y - 5, 1 - x]
sage: vertical_lines_in_braidmon(flist)
[1, 3]
sage: flist += [x * y - 1]
sage: vertical_lines_in_braidmon(flist)
[]
sage: vertical_lines_in_braidmon([])
[]

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import vertical_lines_in_braidmon
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> flist = [x**Integer(2) - y**Integer(3), x, x + Integer(3) * y - Integer(5), Integer(1) - x]
>>> vertical_lines_in_braidmon(flist)
[1, 3]
>>> flist += [x * y - Integer(1)]
>>> vertical_lines_in_braidmon(flist)
[]
>>> vertical_lines_in_braidmon([])
[]
sage.schemes.curves.zariski_vankampen.voronoi_cells(V, vertical_lines=frozenset())[source]#

Compute the graph, the boundary graph, a base point, a positive orientation of the boundary graph, and the dual graph of a corrected Voronoi diagram.

INPUT:

  • V – a corrected Voronoi diagram

  • vertical_lines – frozenset (default: frozenset()); indices of the vertical lines

OUTPUT:

  • G – the graph of the 1-skeleton of V

  • E – the subgraph of the boundary

  • p – a vertex in E

  • EC – a list of vertices (representing a counterclockwise orientation of E) with identical first and last elements)

  • DG – the dual graph of V, where the vertices are labelled by the compact regions of V and the edges by their dual edges.

  • vertical_regions – dictionary for the regions associated with vertical lines

EXAMPLES:

sage: from sage.schemes.curves.zariski_vankampen import corrected_voronoi_diagram, voronoi_cells
sage: points = (2, I, 0.000001, 0, 0.000001*I)
sage: V = corrected_voronoi_diagram(points)
sage: G, E, p, EC, DG, VR = voronoi_cells(V, vertical_lines=frozenset((1,)))
sage: Gv = G.vertices(sort=True)
sage: Ge = G.edges(sort=True)
sage: len(Gv), len(Ge)
(12, 16)
sage: Ev = E.vertices(sort=True); Ev
[A vertex at (-4, 4),
A vertex at (-49000001/14000000, 1000001/2000000),
A vertex at (-7/2, -7/2),
A vertex at (-7/2, 1/2000000),
A vertex at (1/2000000, -7/2),
A vertex at (2000001/2000000, -24500001/7000000),
A vertex at (11/4, 4),
A vertex at (9/2, -9/2),
A vertex at (9/2, 9/2)]
sage: Ev.index(p)
7
sage: EC
(A vertex at (9/2, -9/2),
A vertex at (9/2, 9/2),
A vertex at (11/4, 4),
A vertex at (-4, 4),
A vertex at (-49000001/14000000, 1000001/2000000),
A vertex at (-7/2, 1/2000000),
A vertex at (-7/2, -7/2),
A vertex at (1/2000000, -7/2),
A vertex at (2000001/2000000, -24500001/7000000),
A vertex at (9/2, -9/2))
sage: len(DG.vertices(sort=True)), len(DG.edges(sort=True))
(5, 7)
sage: edg = DG.edges(sort=True)[0]; edg
((0,
  (A vertex at (9/2, -9/2),
   A vertex at (9/2, 9/2),
   A vertex at (11/4, 4),
   A vertex at (2000001/2000000, 500001/1000000),
   A vertex at (2000001/2000000, -24500001/7000000),
   A vertex at (9/2, -9/2))),
 (1,
  (A vertex at (-49000001/14000000, 1000001/2000000),
   A vertex at (1000001/2000000, 1000001/2000000),
   A vertex at (2000001/2000000, 500001/1000000),
   A vertex at (11/4, 4),
   A vertex at (-4, 4),
   A vertex at (-49000001/14000000, 1000001/2000000))),
 (A vertex at (2000001/2000000, 500001/1000000), A vertex at (11/4, 4), None))
sage: edg[-1] in Ge
True
sage: VR
{1: (A vertex at (-49000001/14000000, 1000001/2000000),
     A vertex at (1000001/2000000, 1000001/2000000),
     A vertex at (2000001/2000000, 500001/1000000),
     A vertex at (11/4, 4),
     A vertex at (-4, 4),
     A vertex at (-49000001/14000000, 1000001/2000000))}

>>> from sage.all import *
>>> from sage.schemes.curves.zariski_vankampen import corrected_voronoi_diagram, voronoi_cells
>>> points = (Integer(2), I, RealNumber('0.000001'), Integer(0), RealNumber('0.000001')*I)
>>> V = corrected_voronoi_diagram(points)
>>> G, E, p, EC, DG, VR = voronoi_cells(V, vertical_lines=frozenset((Integer(1),)))
>>> Gv = G.vertices(sort=True)
>>> Ge = G.edges(sort=True)
>>> len(Gv), len(Ge)
(12, 16)
>>> Ev = E.vertices(sort=True); Ev
[A vertex at (-4, 4),
A vertex at (-49000001/14000000, 1000001/2000000),
A vertex at (-7/2, -7/2),
A vertex at (-7/2, 1/2000000),
A vertex at (1/2000000, -7/2),
A vertex at (2000001/2000000, -24500001/7000000),
A vertex at (11/4, 4),
A vertex at (9/2, -9/2),
A vertex at (9/2, 9/2)]
>>> Ev.index(p)
7
>>> EC
(A vertex at (9/2, -9/2),
A vertex at (9/2, 9/2),
A vertex at (11/4, 4),
A vertex at (-4, 4),
A vertex at (-49000001/14000000, 1000001/2000000),
A vertex at (-7/2, 1/2000000),
A vertex at (-7/2, -7/2),
A vertex at (1/2000000, -7/2),
A vertex at (2000001/2000000, -24500001/7000000),
A vertex at (9/2, -9/2))
>>> len(DG.vertices(sort=True)), len(DG.edges(sort=True))
(5, 7)
>>> edg = DG.edges(sort=True)[Integer(0)]; edg
((0,
  (A vertex at (9/2, -9/2),
   A vertex at (9/2, 9/2),
   A vertex at (11/4, 4),
   A vertex at (2000001/2000000, 500001/1000000),
   A vertex at (2000001/2000000, -24500001/7000000),
   A vertex at (9/2, -9/2))),
 (1,
  (A vertex at (-49000001/14000000, 1000001/2000000),
   A vertex at (1000001/2000000, 1000001/2000000),
   A vertex at (2000001/2000000, 500001/1000000),
   A vertex at (11/4, 4),
   A vertex at (-4, 4),
   A vertex at (-49000001/14000000, 1000001/2000000))),
 (A vertex at (2000001/2000000, 500001/1000000), A vertex at (11/4, 4), None))
>>> edg[-Integer(1)] in Ge
True
>>> VR
{1: (A vertex at (-49000001/14000000, 1000001/2000000),
     A vertex at (1000001/2000000, 1000001/2000000),
     A vertex at (2000001/2000000, 500001/1000000),
     A vertex at (11/4, 4),
     A vertex at (-4, 4),
     A vertex at (-49000001/14000000, 1000001/2000000))}