sagemathgh-38794: convert magma power series rings
    
this is trying to allow the conversion of magma's series rings

also refactors the conversion of polynomial rings

This needs to be tested, if you have magma.

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#38794
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/ext_data/magma/sage/basic.m

@@ -8,7 +8,7 @@ function PreparseElts(R)
 end function;
 
 intrinsic Sage(X::.) -> MonStgElt, BoolElt
-{Default way to convert a Magma object to Sage if we haven't
+{Default way to convert a Magma object to Sage if we have not
 written anything better.}
     return Sprintf("%o", X), true;
 end intrinsic;
@@ -153,16 +153,20 @@ intrinsic SageNamesHelper(X::.) -> MonStgElt
 {}
   /* XXX */
   i := NumberOfNames(X);
-  if i ge 2 then
-      return (&* [ Sprintf("%o, ", X.j) : j in [ 1..i-1 ] ]) * Sprintf("%o", X.i);
+  if "$" in Sprint(X.i) then
+    /* unnamed variables */
+    return "(" * (&* [ Sprintf("'x%o', ", j) : j in [ 1..i ] ]) * ")";
   else
-      return Sprintf("%o", X.i);
-  end if;
+    /* named variables */
+    return "(" * (&* [ Sprintf("'%o'.replace('.', ''), ", X.j) : j in [ 1..i ] ]) * ")";
+
+end if;
 end intrinsic;
 
 intrinsic Sage(X::RngUPol) -> MonStgElt, BoolElt
 {}
-  return Sprintf("%o['%o'.replace('$.', 'x').replace('.', '')]", Sage(BaseRing(X)), SageNamesHelper(X)), false;
+  txt := "PolynomialRing(%o, %o)";
+  return Sprintf(txt, Sage(BaseRing(X)), SageNamesHelper(X)), false;
 end intrinsic;
 
 intrinsic Sage(X::RngUPolElt) -> MonStgElt, BoolElt
@@ -173,7 +177,8 @@ intrinsic Sage(X::RngUPolElt) -> MonStgElt, BoolElt
 
 intrinsic Sage(X::RngMPol) -> MonStgElt, BoolElt
 {}
-  return Sprintf("%o['%o'.replace('$.', 'x').replace('.', '')]", Sage(BaseRing(X)), SageNamesHelper(X)), false;
+  txt := "PolynomialRing(%o, %o)";
+  return Sprintf(txt, Sage(BaseRing(X)), SageNamesHelper(X)), false;
 end intrinsic;
 
 intrinsic Sage(X::RngMPolElt) -> MonStgElt, BoolElt
@@ -187,9 +192,16 @@ intrinsic Sage(X::RngMPolElt) -> MonStgElt, BoolElt
 
 intrinsic Sage(K::FldNum) -> MonStgElt, BoolElt
 {}
-    names := [Sprintf("'%o'.replace('$.', 'a').replace('.', '')", a) : a in GeneratorsSequence(K)];
-    polynomials := DefiningPolynomial(K);
-    return Sprintf("NumberField(%o, %o)", Sage(polynomials), names), false;
+  gens := GeneratorsSequence(K);
+  if "$" in Sprint(gens[1]) then
+    /* unnamed variables */
+    names := "(" * (&* [ Sprintf("'a%o', ", j) : j in [ 1..#gens ] ]) * ")";
+  else
+    /* named variables */
+    names := "(" * (&* [ Sprintf("'%o'.replace('.', ''), ", a) : a in gens]) * ")";
+  end if;
+  polynomials := DefiningPolynomial(K);
+  return Sprintf("NumberField(%o, %o)", Sage(polynomials), names), false;
 end intrinsic;
 
 intrinsic Sage(A::FldNumElt) -> MonStgElt, BoolElt
@@ -264,3 +276,26 @@ intrinsic Sage(X::ModTupRngElt) -> MonStgElt, BoolElt
 {}
     return Sprintf("%o(%o)", Sage(Parent(X)), Sage(ElementToSequence(X))), true;
 end intrinsic;
+
+/* Power series rings */
+
+intrinsic Sage(X::RngSerPow) -> MonStgElt, BoolElt
+{}
+  txt := "PowerSeriesRing(%o, %o)";
+  var := Sprintf("['%o']", X.1);
+  return Sprintf(txt, Sage(BaseRing(X)), var), false;
+end intrinsic;
+
+intrinsic Sage(X::RngSerLaur) -> MonStgElt, BoolElt
+{}
+  txt := "LaurentSeriesRing(%o, %o)";
+  var := Sprintf("['%o']", X.1);
+  return Sprintf(txt, Sage(BaseRing(X)), var), false;
+end intrinsic;
+
+intrinsic Sage(X::RngSerPuis) -> MonStgElt, BoolElt
+{}
+  txt := "PuiseuxSeriesRing(%o, %o)";
+  var := Sprintf("['%o']", X.1);
+  return Sprintf(txt, Sage(BaseRing(X)), var), false;
+end intrinsic;

src/sage/interfaces/magma.py

@@ -10,7 +10,7 @@
    You must have Magma installed on your
    computer for this interface to work. Magma is not free, so it is
    not included with Sage, but you can obtain it from
-   http://magma.maths.usyd.edu.au/.
+   https://magma.maths.usyd.edu.au/.
 
 The Magma interface offers three pieces of functionality:
 
@@ -214,34 +214,35 @@
 #
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
-from __future__ import annotations
+from pathlib import Path
 import re
 import sys
 import os
 
 from sage.structure.parent import Parent
 from .expect import Expect, ExpectElement, ExpectFunction, FunctionElement
-PROMPT = ">>>"
-
-SAGE_REF = "_sage_ref"
-SAGE_REF_RE = re.compile(r'%s\d+' % SAGE_REF)
-
 from sage.env import SAGE_EXTCODE, DOT_SAGE
 import sage.misc.misc
 import sage.misc.sage_eval
 import sage.interfaces.abc
 from sage.interfaces.tab_completion import ExtraTabCompletion
 from sage.misc.instancedoc import instancedoc
 
+PROMPT = ">>>"
+
+SAGE_REF = "_sage_ref"
+SAGE_REF_RE = re.compile(r'%s\d+' % SAGE_REF)
+
 INTRINSIC_CACHE = '%s/magma_intrinsic_cache.sobj' % DOT_SAGE
 EXTCODE_DIR = None
 
 
-def extcode_dir(iface=None):
+def extcode_dir(iface=None) -> str:
     """
-    Return directory that contains all the Magma extcode.  This is put
-    in a writable directory owned by the user, since when attached,
-    Magma has to write sig and lck files.
+    Return directory that contains all the Magma extcode.
+
+    This is put in a writable directory owned by the user, since when
+    attached, Magma has to write sig and lck files.
 
     EXAMPLES::
 
@@ -413,7 +414,7 @@ def __reduce__(self):
         """
         return reduce_load_Magma, tuple([])
 
-    def _read_in_file_command(self, filename):
+    def _read_in_file_command(self, filename) -> str:
         """
         Return the command in Magma that reads in the contents of the given
         file.
@@ -433,7 +434,7 @@ def _read_in_file_command(self, filename):
         """
         return 'load "%s";' % filename
 
-    def _post_process_from_file(self, s):
+    def _post_process_from_file(self, s) -> str:
         r"""
         Used internally in the Magma interface to post-process the result
         of evaluating a string using a file. For Magma what this does is
@@ -494,7 +495,7 @@ def __getattr__(self, attrname):
             raise AttributeError
         return MagmaFunction(self, attrname)
 
-    def eval(self, x, strip=True, **kwds):
+    def eval(self, x, strip=True, **kwds) -> str:
         """
         Evaluate the given block x of code in Magma and return the output
         as a string.
@@ -553,7 +554,7 @@ def eval(self, x, strip=True, **kwds):
             raise RuntimeError("Error evaluating Magma code.\nIN:%s\nOUT:%s" % (x, ans))
         return ans
 
-    def _preparse(self, s):
+    def _preparse(self, s) -> str:
         """
         All input gets preparsed by calling this function before it gets evaluated.
 
@@ -578,7 +579,7 @@ def _preparse(self, s):
             pass
         return s
 
-    def _start(self):
+    def _start(self) -> None:
         """
         Initialize a Magma interface instance. This involves (1) setting up
         an obfuscated prompt, and (2) attaching the MAGMA_SPEC file (see
@@ -619,7 +620,7 @@ def set(self, var, value):
         if out.lower().find("error") != -1:
             raise TypeError("Error executing Magma code:\n%s" % out)
 
-    def get(self, var):
+    def get(self, var) -> str:
         """
         Get the value of the variable var.
 
@@ -655,7 +656,7 @@ def objgens(self, value, gens):
 
             sage: R = magma.objgens('PolynomialRing(Rationals(),2)', 'alpha,beta')    # optional - magma
             sage: R.gens()          # optional - magma
-            [alpha, beta]
+            (alpha, beta)
 
         Because of how Magma works you can use this to change the variable
         names of the generators of an object::
@@ -911,16 +912,15 @@ def cputime(self, t=None):
             sage: # optional - magma
             sage: type(magma.cputime())
             <... 'float'>
-            sage: magma.cputime()
+            sage: magma.cputime()  # random
             1.9399999999999999
             sage: t = magma.cputime()
-            sage: magma.cputime(t)
+            sage: magma.cputime(t)  # random
             0.02
         """
         if t:
             return float(self.eval('Cputime(%s)' % t))
-        else:
-            return float(self.eval('Cputime()'))
+        return float(self.eval('Cputime()'))
 
     def chdir(self, dir):
         """
@@ -1003,7 +1003,7 @@ def load(self, filename):
 
         Loading a file in Magma makes all the functions and procedures in
         the file available. The file should not contain any intrinsics (or
-        you'll get errors). It also runs code in the file, which can
+        you will get errors). It also runs code in the file, which can
         produce output.
 
         INPUT:
@@ -1018,14 +1018,15 @@ def load(self, filename):
             sage: with NTF(mode='w+t', suffix='.m') as f:  # optional - magma
             ....:     _ = f.write('function f(n) return n^2; end function;\nprint "hi";')
             ....:     print(magma.load(f.name))
-            Loading ".../a.m"
+            Loading "....m"
             hi
             sage: magma('f(12)')  # optional - magma
             144
         """
-        return self.eval('load "%s"' % filename)
+        p = Path(filename)
+        return self.eval('load "%s"' % p.absolute())
 
-    def _next_var_name(self):
+    def _next_var_name(self) -> str:
         """
         Return the next available variable name in Magma.
 
@@ -1234,7 +1235,7 @@ def bar_call(self, left, name, gens, nvals=1):
         magma = self
         # coerce each arg to be a Magma element
         if isinstance(gens, (list, tuple)):
-            gens = (magma(z) for z in gens)
+            gens = [magma(z) for z in gens]
             # make comma separated list of names (in Magma) of each of the gens
             v = ', '.join(w.name() for w in gens)
         else:
@@ -1880,10 +1881,11 @@ def __getattr__(self, attrname):
 
     def _sage_(self):
         """
-        Return Sage version of this object. Use self.sage() to get the Sage
-        version.
+        Return Sage version of this object.
+
+        Use self.sage() to get the Sage version.
 
-        Edit src/ext/magma/sage/basic.m to add functionality.
+        Edit ``src/sage/ext_data/magma/sage/basic.m`` to add functionality.
 
         EXAMPLES: Enumerated Sets::
 
@@ -2070,14 +2072,14 @@ def AssignNames(self, names):
 
     def gen(self, n):
         """
-        Return the `n`-th generator of this Magma element. Note that
-        generators are 1-based in Magma rather than 0-based!
+        Return the `n`-th generator of this Magma element.
+
+        Note that generators are 1-based in Magma rather than 0-based!
 
         INPUT:
 
         - ``n`` -- *positive* integer
 
-
         OUTPUT: :class:`MagmaElement`
 
         EXAMPLES::
@@ -2100,7 +2102,7 @@ def gen(self, n):
             sage: m.gen(4)                # optional -- magma
             Traceback (most recent call last):
             ...
-            IndexError: list index out of range
+            IndexError: tuple index out of range
         """
         if n <= 0:
             raise IndexError("index must be positive since Magma indexes are 1-based")
@@ -2112,7 +2114,7 @@ def gens(self) -> tuple:
 
         If ``self`` is named X in Magma, this function evaluates X.1, X.2,
         etc., in Magma until an error occurs. It then returns a Sage tuple
-        of the resulting X.i. Note - I don't think there is a Magma command
+        of the resulting X.i. Note - I do not think there is a Magma command
         that returns the list of valid X.i. There are numerous ad hoc
         functions for various classes but nothing systematic. This function
         gets around that problem. Again, this is something that should
@@ -2294,15 +2296,15 @@ def _polynomial_(self, R):
             sage: R.<x> = QQ[]
             sage: f = magma(x^2 + 2/3*x + 5)                 # optional - magma
             sage: f                                          # optional - magma
-            t^2 + 2/3*t + 5
+            x^2 + 2/3*x + 5
             sage: f.Type()                                   # optional - magma
             RngUPolElt
             sage: f._polynomial_(R)                          # optional - magma
             x^2 + 2/3*x + 5
         """
         return R(list(self.Eltseq()))
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return latex representation of ``self``.
 
@@ -2854,6 +2856,7 @@ def write(self, s):
             sage: P.<x,y,z> = GF(32003)[]
             sage: I = sage.rings.ideal.Katsura(P)
             sage: _ = I.groebner_basis('magma',prot=True) # indirect doctest, optional - magma
+            ...
             ********************
             FAUGERE F4 ALGORITHM
             ********************

src/sage/matrix/matrix_mod2_dense.pyx

@@ -1393,14 +1393,15 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
 
     def _magma_init_(self, magma):
         """
-        Return a string of ``self`` in ``Magma`` form. Does not return
-        ``Magma`` object but string.
+        Return a string of ``self`` in ``Magma`` form.
+
+        This does not return a ``Magma`` object but a string.
 
         EXAMPLES::
 
             sage: A = random_matrix(GF(2),3,3)
             sage: A._magma_init_(magma)                             # optional - magma
-            'Matrix(GF(2),3,3,StringToIntegerSequence("0 1 0 0 1 1 0 0 0"))'
+            'Matrix(GF(2),3,3,StringToIntegerSequence("..."))'
             sage: A = random_matrix(GF(2),100,100)
             sage: B = random_matrix(GF(2),100,100)
             sage: magma(A*B) == magma(A) * magma(B)                 # optional - magma

src/sage/modular/btquotients/btquotient.py

@@ -42,6 +42,7 @@
 from collections import deque
 
 from sage.arith.misc import gcd, xgcd, kronecker_symbol, fundamental_discriminant
+from sage.interfaces.magma import magma
 from sage.matrix.constructor import Matrix
 from sage.matrix.matrix_space import MatrixSpace
 from sage.misc.cachefunc import cached_method
@@ -2307,7 +2308,7 @@ def get_extra_embedding_matrices(self):
             [1 0 2 0]
             [0 0 2 0]
             [0 0 0 0]
-            [1 0 2 2]
+            [1 2 2 0]
             ]
         """
         if not self._use_magma or len(self._extra_level) == 0:
@@ -2700,7 +2701,7 @@ def get_splitting_field(self):
 
             sage: X = BruhatTitsQuotient(5,11,use_magma=True) # optional - magma
             sage: X.get_splitting_field() # optional - magma
-            Number Field in a with defining polynomial X1^2 + 11
+            Number Field in a with defining polynomial x^2 + 11
         """
         if not self._use_magma:
             raise NotImplementedError('Sage does not know yet how to work with the kind of orders that you are trying to use. Try installing Magma first and set it up so that Sage can use it.')