gh-35086: move single_valued method of MZV to auxiliary F ring
    
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This is moving a little piece of code from MZV to a more proper place
(auxiliary ring).

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URL: #35086
Reported by: Frédéric Chapoton
Reviewer(s): Vincent Delecroix

Author: Release Manager

src/doc/en/reference/references/index.rst

@@ -1184,6 +1184,9 @@ REFERENCES:
              moduli spaces to Feynman integrals*, in Contemporary Mathematics
              vol 539, pages 27-52, 2011.
 
+.. [Bro2013] Francis Brown, *Single-valued motivic periods and multiple zeta
+             values*, Forum Math. Sigma 2 (2014), :doi:`10.1017/fms.2014.18`.
+
 .. [Bro2016] \A.E. Brouwer,
              Personal communication, 2016.
 

src/sage/modular/multiple_zeta.py

@@ -1083,9 +1083,15 @@ def iterated(self):
             return self.parent().iterated(self)
 
         def single_valued(self):
-            """
+            r"""
             Return the single-valued version of ``self``.
 
+            This is the projection map onto the sub-algebra of
+            single-valued motivic multiple zeta values, as defined by
+            F. Brown in [Bro2013]_.
+
+            This morphism of algebras sends in particular `\zeta(2)` to `0`.
+
             EXAMPLES::
 
                 sage: M = Multizetas(QQ)
@@ -1108,13 +1114,7 @@ def single_valued(self):
                 sage: Z(5,3).single_valued() == 14*Z(3)*Z(5)
                 True
             """
-            phi_im = self.phi()
-            zin = phi_im.parent()
-            phi_no_f2 = phi_im.without_f2()
-            sv = zin.sum_of_terms(((0, w), cf)
-                                  for (a, b), cf in phi_no_f2.coproduct()
-                                  for w in shuffle(a[1], b[1].reversal(), False))
-            return rho_inverse(sv)
+            return rho_inverse(self.phi().single_valued())
 
         def simplify(self):
             """

src/sage/modular/multiple_zeta_F_algebra.py

@@ -31,6 +31,7 @@
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.combinat.words.words import Words
 from sage.combinat.words.finite_word import FiniteWord_class
+from sage.combinat.words.shuffle_product import ShuffleProduct_w1w2 as shuffle
 from sage.misc.cachefunc import cached_method
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.sets.integer_range import IntegerRange
@@ -715,9 +716,30 @@ def without_f2(self):
 
                 sage: from sage.modular.multiple_zeta_F_algebra import F_algebra
                 sage: F = F_algebra(QQ)
-                sage: t = 4*F("35")+F("27")
+                sage: t = 4 * F("35") + F("27")
                 sage: t.without_f2()
                 4*f3f5
             """
             F = self.parent()
             return F._from_dict({(0, w): cf for (p, w), cf in self if not p})
+
+        def single_valued(self):
+            """
+            Return the single-valued version of ``self``.
+
+            EXAMPLES::
+
+                sage: from sage.modular.multiple_zeta_F_algebra import F_algebra
+                sage: F = F_algebra(QQ)
+                sage: t = 4 * F("2") + F("3")
+                sage: t.single_valued()
+                2*f3
+                sage: t = 4 * F("35") + F("27")
+                sage: t.single_valued()
+                8*f3f5 + 8*f5f3
+            """
+            F = self.parent()
+            no_f2 = self.without_f2()
+            return F.sum_of_terms(((0, w), cf)
+                                  for (a, b), cf in no_f2.coproduct()
+                                  for w in shuffle(a[1], b[1].reversal(), False))