sagemathgh-39301: Using a new functions in libbraiding in zariski_van…

…kampen

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The file `zariski_vankampen` used some `libbraiding` functions in order
to compute the relations of the  group obtained applying Zariski-van
Kampen method. The use of the `super_summit_set` allowed to end
computations which did not finish with a straightforward approach. But,
when one works with large braid groups, the mehtod `super_summit_set`
can be very long, and running along its output is even longer. The new
`libbraiding` version includes the method `super_summit_set_element`
which simplifies and fasten the computations. The main changes are in
method `conjugate_positive_form`

### 📝 Checklist

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- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [X] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#39301
Reported by: Enrique Manuel Artal Bartolo
Reviewer(s): Travis Scrimshaw

build/pkgs/configure/checksums.ini

@@ -1,3 +1,3 @@
 tarball=configure-VERSION.tar.gz
-sha1=852d0d200a6a73aa5ddb9e00874cbe4a61c211e9
-sha256=c4b089d90850dfdf15b905f66e4f6a0d961b96eb0663d8603beaff1a9efb2cbe
+sha1=1eddcd428863dd3f9a9b59ac701e63d9b718748d
+sha256=bb763394dfaa0ba132b9e8055eaca3d3fb0cae9135403582eff41491d40248e5

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-a2ba1f943f88775218c385efe55509c4548d1b44
+0cb2721d2a823ab7042811c7fca5752c54248e67

src/sage/geometry/hyperplane_arrangement/ordered_arrangement.py

@@ -394,7 +394,6 @@ def projective_fundamental_group(self):
             < x0, x1, x2, x3, x4 | x4^-1*x3^-1*x2^-1*x3*x4*x0*x2*x0^-1,
                                    x4^-1*x2^-1*x4*x2, x4^-1*x1^-1*x0^-1*x1*x4*x0,
                                    x4^-1*x1^-1*x0^-1*x4*x0*x1,
-                                   x4^-1*x1^-1*x3*x0*x1*x3^-1*x2^-1*x4*x0^-1*x2,
                                    x3^-1*x2^-1*x1^-1*x0^-1*x3*x0*x1*x2,
                                    x3^-1*x1^-1*x3*x1 >
             sage: G3.abelian_invariants()
@@ -406,9 +405,7 @@ def projective_fundamental_group(self):
             < x0, x1, x2, x3, x4 | x4^-1*x3^-1*x2^-1*x3*x4*x0*x2*x0^-1,
                                    x4^-1*x2^-1*x4*x2, x4^-1*x1^-1*x0^-1*x1*x4*x0,
                                    x4^-1*x1^-1*x0^-1*x4*x0*x1,
-                                   x4^-1*x1^-1*x3*x0*x1*x3^-1*x2^-1*x4*x0^-1*x2,
-                                   x3^-1*x2^-1*x1^-1*x0^-1*x3*x0*x1*x2,
-                                   x3^-1*x1^-1*x3*x1 >
+                                   x3^-1*x2^-1*x1^-1*x0^-1*x3*x0*x1*x2, x3^-1*x1^-1*x3*x1 >
             sage: G4.abelian_invariants()
             (0, 0, 0, 0, 0)
 

src/sage/schemes/curves/plane_curve_arrangement.py

@@ -513,7 +513,7 @@ def fundamental_group(self, simplified=True, vertical=True,
             {0: [x1, x2], 1: [x0], 2: [x3], 3: [x3^-1*x2^-1*x1^-1*x0^-1]}
             sage: A.fundamental_group(vertical=False)
             Finitely presented group
-            < x0, x1, x2 | x2^-1*x1^-1*x2*x1, x1*x0*x1^-1*x0^-1, (x0*x2)^2*(x0^-1*x2^-1)^2 >
+            < x0, x1, x2 | x2*x1^-1*x2^-1*x1, x1*x0*x1^-1*x0^-1, (x0*x2)^2*(x0^-1*x2^-1)^2 >
             sage: A.meridians(vertical=False)
             {0: [x2, x0*x2*x0^-1], 1: [x1], 2: [x0], 3: [x0*x2^-1*x0^-1*x2^-1*x1^-1*x0^-1]}
             sage: G = A.fundamental_group(simplified=False, vertical=False)
@@ -834,7 +834,7 @@ def fundamental_group(self, simplified=True):
             sage: A.fundamental_group().sorted_presentation()
             Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >
             sage: A.meridians()
-            {0: [x1], 1: [x0], 2: [x1^-1*x0^-1*x1^-1]}
+            {0: [x1], 1: [x0], 2: [x0^-1*x1^-2]}
             sage: G = A.fundamental_group(simplified=False)
             sage: G.sorted_presentation()
             Finitely presented group
@@ -945,7 +945,7 @@ def meridians(self, simplified=True):
             sage: A.fundamental_group().sorted_presentation()
             Finitely presented group < x0, x1 | x1^-1*x0^-1*x1*x0 >
             sage: A.meridians()
-            {0: [x1], 1: [x0], 2: [x1^-1*x0^-1*x1^-1]}
+            {0: [x1], 1: [x0], 2: [x0^-1*x1^-2]}
             sage: A = H(y^2 + x*z, z, x)
             sage: A.fundamental_group()
             Finitely presented group < x0, x1 | (x1*x0)^2*(x1^-1*x0^-1)^2 >

src/sage/schemes/curves/projective_curve.py

@@ -1775,8 +1775,9 @@ def fundamental_group(self):
             sage: C = P.curve(x^2*z - y^3)
             sage: C.fundamental_group()                                 # needs sirocco
             Finitely presented group < x0 | x0^3 >
-            sage: P.curve(z*(x^2*z - y^3)).fundamental_group()          # needs sirocco
-            Finitely presented group < x0, x1 | x1*x0*x1*x0^-1*x1^-1*x0^-1 >
+            sage: g = P.curve(z*(x^2*z - y^3)).fundamental_group()      # needs sirocco
+            sage: g.sorted_presentation()                               # needs sirocco
+            Finitely presented group < x0, x1 | x1^-1*x0^-1*x1^-1*x0*x1*x0 >
 
         In the case of number fields, they need to have an embedding
         into the algebraic field::

src/sage/schemes/curves/zariski_vankampen.py

@@ -1416,7 +1416,7 @@ def conjugate_positive_form(braid):
 
     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_{1}^{i},\dots,\gamma_{n_i}^{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`.
@@ -1426,51 +1426,56 @@ def conjugate_positive_form(braid):
         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: cpf = conjugate_positive_form(t); cpf
+        [[(s0*s1)^2, [s0*s2*s1*s0]]]
+        sage: t == prod(prod(b) * a / prod(b) for a, b in cpf)
+        True
         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
+        [[s0^2, [s0*s1*s2*s1*s3*s2*s1*s0]], [s3*s2, [s0*s1*s2*s1*s3*s2*s1*s0]]]
+        sage: t == prod(prod(b) * a / prod(b) for a, b in L)
         True
         sage: s1 = B.gen(1)^3
         sage: conjugate_positive_form(s1)
         [[s1^3, []]]
     """
     B = braid.parent()
     d = B.strands()
-    braid1 = braid.super_summit_set()[0]
-    L1 = braid1.Tietze()
-    sg0 = braid.conjugating_braid(braid1)
-    gns = set(L1)
-    cuts = [j for j in range(d + 1) if j not in gns]
-    blocks = []
-    for i in range(len(cuts) - 1):
-        block = [j for j in L1 if cuts[i] < j < cuts[i + 1]]
-        if block:
-            blocks.append(block)
+    rnf = rightnormalform(braid)
+    ex = rnf[-1][0]
+    if ex >= 0:
+        A1 = [B(a) for a in rnf[:-1]]
+        braid1 = prod(A1, B.delta() ** ex)
+        sg0 = B.one()
+    else:
+        braid1, sg0 = braid.super_summit_set_element()
+    if ex > 0:
+        blocks = [list(braid1.Tietze())]
+    else:
+        L1 = braid1.Tietze()
+        gns = set(L1)
+        cuts = [j for j in range(d + 1) if j not in gns]
+        blocks = []
+        for i in range(len(cuts) - 1):
+            block = [j for j in L1 if cuts[i] < j < cuts[i + 1]]
+            if block:
+                blocks.append(block)
     shorts = []
     for a in blocks:
-        A = B(a).super_summit_set()
-        res = None
-        for tau in A:
-            sg = (sg0 * B(a) / sg0).conjugating_braid(tau)
+        if sg0 == B.one():
+            res0 = [B(a), []]
+        else:
+            bra = sg0 * B(a) / sg0
+            br1, sg = bra.super_summit_set_element()
             A1 = rightnormalform(sg)
             par = A1[-1][0] % 2
-            A1 = [B(a) for a in A1[:-1]]
-            b = prod(A1, B.one())
-            b1 = len(b.Tietze()) / (len(A1) + 1)
-            if res is None or b1 < res[3]:
-                res = [tau, A1, par, b1]
-        if res[2] == 1:
-            r0 = res[0].Tietze()
-            res[0] = B([i.sign() * (d - abs(i)) for i in r0])
-        res0 = res[:2]
+            A1 = [B(a0) for a0 in A1[:-1]]
+            res = [br1, A1, par]
+            if res[2]:
+                r0 = res[0].Tietze()
+                res[0] = B([d - i for i in r0])
+            res0 = res[:2]
         shorts.append(res0)
     return shorts
 
@@ -1596,9 +1601,10 @@ def fundamental_group_from_braid_mon(bm, degree=None,
         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
+        sage: g = fundamental_group_from_braid_mon(bm, projective=True)        # needs sirocco
+        sage: g.sorted_presentation()                                          # needs sirocco
         Finitely presented group
-        < x1, x3 | x3^2*x1^2, x1^-1*x3^-1*x1*x3^-1*x1^-1*x3^-1 >
+        < x0, x1 | x1^-2*x0^-2, x1^-1*(x0^-1*x1)^2*x0 >
         sage: print(g.order(), g.abelian_invariants())                         # needs sirocco
         12 (4,)
         sage: B2 = BraidGroup(2)
@@ -1697,8 +1703,8 @@ def fundamental_group(f, simplified=True, projective=False, puiseux=True):
         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).sorted_presentation()
+        Finitely presented group < x0, x1 | x1^-1*x0^-1*x1^-1*x0*x1*x0 >
         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,
@@ -1859,7 +1865,7 @@ def fundamental_group_arrangement(flist, simplified=True, projective=False,
         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 >
+                           x3^-1*x1^-1*x0^-1*x3*x0*x1, 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)