Trac #28571: pass 'macaulay2' as argument to '_macaulay2_init_()'

This gives access to the Macaulay2 interpreter in the
`_macaulay2_init_()` function, so it can also return `Macaulay2Elements`
rather than just strings. The `_macaulay2_init_()` function is used for
initializing Macaulay2 elements and is usually called by `_macaulay2_()`
which then takes care of caching.

The magma interface also does this, but it had not been correctly set up
until now.

URL: https://trac.sagemath.org/28571
Reported by: gh-mwageringel
Ticket author(s): Markus Wageringel
Reviewer(s): Franco Saliola

Author: Release Manager

src/sage/rings/finite_rings/finite_field_base.pyx

@@ -243,7 +243,7 @@ cdef class FiniteField(Field):
         s = "ext<%s|%s>"%(B._magma_init_(magma),p._magma_init_(magma))
         return magma._with_names(s, self.variable_names())
 
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         """
         Returns the string representation of ``self`` that Macaulay2 can
         understand.

src/sage/rings/finite_rings/integer_mod_ring.py

@@ -480,7 +480,7 @@ def __init__(self, order, cache=None, category=None):
         self._zero_element = integer_mod.IntegerMod(self, 0)
         self._one_element = integer_mod.IntegerMod(self, 1)
 
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         """
         EXAMPLES::
 

src/sage/rings/integer_ring.pyx

@@ -1452,7 +1452,7 @@ cdef class IntegerRing_class(PrincipalIdealDomain):
         """
         return 'IntegerRing()'
 
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         """
         Return a macaulay2 representation of ``self``.
 

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -1189,10 +1189,10 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
             R._check_valid()
             return R
         except (AttributeError, ValueError):
-            self.__macaulay2 = macaulay2(self._macaulay2_init_())
+            self.__macaulay2 = macaulay2(self._macaulay2_init_(macaulay2))
         return self.__macaulay2
 
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         """
         EXAMPLES::
 

src/sage/rings/polynomial/multi_polynomial_ring.py

@@ -83,7 +83,7 @@ class MPolynomialRing_macaulay2_repr:
     """
     A mixin class for polynomial rings that support conversion to Macaulay2.
     """
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         """
         EXAMPLES::
 

src/sage/rings/polynomial/polynomial_ring.py

@@ -913,7 +913,7 @@ def _sage_input_(self, sib, coerced):
         return sib.parent_with_gens(self, sie, self.variable_names(), 'R',
                                     gens_syntax=gens_syntax)
 
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         """
         EXAMPLES::
 

src/sage/rings/rational_field.py

@@ -1427,7 +1427,7 @@ def _magma_init_(self, magma):
         """
         return 'RationalField()'
 
-    def _macaulay2_init_(self):
+    def _macaulay2_init_(self, macaulay2=None):
         r"""
         Return the macaulay2 representation of `\QQ`.
 

src/sage/structure/sage_object.pyx

@@ -30,6 +30,10 @@ register_unpickle_override('sage.structure.generators', 'make_list_gens',
 __all__ = ['SageObject']
 
 
+# The _interface_init_ for these interfaces takes the interface as argument
+_interface_init_with_interface = set(['magma', 'macaulay2'])
+
+
 cdef class SageObject:
     """
     Base class for all (user-visible) objects in Sage
@@ -660,7 +664,10 @@ cdef class SageObject:
         nm = I.name()
         init_func = getattr(self, '_%s_init_' % nm, None)
         if init_func is not None:
-            s = init_func()
+            if nm in _interface_init_with_interface:
+                s = init_func(I)
+            else:
+                s = init_func()
         else:
             try:
                 s = self._interface_init_(I)
@@ -831,10 +838,11 @@ cdef class SageObject:
             G = sage.interfaces.macaulay2.macaulay2
         return self._interface_(G)
 
-    def _macaulay2_init_(self):
-        import sage.interfaces.macaulay2
-        I = sage.interfaces.macaulay2.macaulay2
-        return self._interface_init_(I)
+    def _macaulay2_init_(self, macaulay2=None):
+        if macaulay2 is None:
+            import sage.interfaces.macaulay2
+            macaulay2 = sage.interfaces.macaulay2.macaulay2
+        return self._interface_init_(macaulay2)
 
     def _maple_(self, G=None):
         if G is None: