Trac #28591: convert chain complexes from Macaulay2 to Sage

This ticket implements conversion of chain complexes from Macaulay2 to
Sage, for example:
{{{
            sage: R = ZZ['a,b,c']
            sage: C = macaulay2(ideal(R.gens())).resolution()
            sage: unicode_art(C.sage())
                                   ⎛-b  0 -c⎞     ⎛ c⎞
                                   ⎜ a -c  0⎟     ⎜ a⎟
                       (a b c)     ⎝ 0  b  a⎠     ⎝-b⎠
             0 ⟵── C_0 ⟵────── C_1 ⟵───────── C_2 ⟵─── C_3 ⟵── 0
}}}

This ticket also fixes an issue where matrices of size 0×n failed to
convert correctly.

URL: https://trac.sagemath.org/28591
Reported by: gh-mwageringel
Ticket author(s): Markus Wageringel
Reviewer(s): Frédéric Chapoton, Dima Pasechnik

src/sage/interfaces/macaulay2.py

@@ -692,6 +692,8 @@ def help(self, s):
         """
         EXAMPLES::
 
+            sage: macaulay2.help("load")  # optional - macaulay2 - 1st call might be chatty...
+            ...
             sage: macaulay2.help("load")  # optional - macaulay2
             load...
             ****...
@@ -1293,7 +1295,7 @@ def underscore(self, x):
     #Conversion to Sage#
     ####################
     def _sage_(self):
-        """
+        r"""
         EXAMPLES::
 
             sage: macaulay2(ZZ).sage()         # optional - macaulay2, indirect doctest
@@ -1366,7 +1368,7 @@ def _sage_(self):
             sage: m.sage()                  # optional - macaulay2
             'hello'
 
-            sage: macaulay2.needsPackage('"Graphs"');   # optional - macaulay2
+            sage: gg = macaulay2.needsPackage('"Graphs"') # optional - macaulay2
             sage: g = macaulay2.barbellGraph(3)         # optional - macaulay2
             sage: g.sage()                              # optional - macaulay2
             Graph on 6 vertices
@@ -1380,6 +1382,27 @@ def _sage_(self):
             sage: g.sage().edges(labels=False)          # optional - macaulay2
             [(1, 2), (2, 1), (3, 1)]
 
+        Chain complexes and maps of chain complexes can be converted::
+
+            sage: R = ZZ['a,b,c']
+            sage: C = macaulay2(ideal(R.gens())).resolution()  # optional - macaulay2
+            sage: ascii_art(C.sage())                          # optional - macaulay2
+                                      [-b  0 -c]       [ c]
+                                      [ a -c  0]       [ a]
+                        [a b c]       [ 0  b  a]       [-b]
+             0 <-- C_0 <-------- C_1 <----------- C_2 <----- C_3 <-- 0
+            sage: F = C.dot('dd')  # optional - macaulay2
+            sage: G = F.sage()     # optional - macaulay2
+            sage: G.in_degree(2)   # optional - macaulay2
+            [-b  0 -c]
+            [ a -c  0]
+            [ 0  b  a]
+            sage: F.underscore(2).sage() == G.in_degree(2)  # optional - macaulay2
+            True
+            sage: (F^2).sage()     # optional - macaulay2
+            Chain complex morphism:
+              From: Chain complex with at most 4 nonzero terms over Multivariate Polynomial Ring in a, b, c over Integer Ring
+              To:   Chain complex with at most 4 nonzero terms over Multivariate Polynomial Ring in a, b, c over Integer Ring
         """
         repr_str = str(self)
         cls_str = str(self.cls())
@@ -1477,6 +1500,30 @@ def _sage_(self):
                 g = graph_cls(adj_mat, format='adjacency_matrix')
                 g.relabel(self.vertices())
                 return g
+            elif cls_str == "ChainComplex":
+                from sage.homology.chain_complex import ChainComplex
+                ring = self.ring()._sage_()
+                dd = self.dot('dd')
+                degree = dd.degree()._sage_()
+                a = self.min()._sage_()
+                b = self.max()._sage_()
+                matrices = {i: dd.underscore(i)._matrix_(ring)
+                            for i in range(a, b+1)}
+                return ChainComplex(matrices, degree=degree)
+            elif cls_str == "ChainComplexMap":
+                from sage.homology.chain_complex_morphism import ChainComplexMorphism
+                ring = self.ring()._sage_()
+                source = self.source()
+                a = source.min()._sage_()
+                b = source.max()._sage_()
+                degree = self.degree()._sage_()
+                matrices = {i: self.underscore(i)._matrix_(ring)
+                            for i in range(a, b+1)}
+                C = source._sage_()
+                # in Sage, chain complex morphisms are degree-preserving,
+                # so we shift the degrees of the target
+                D = self.target()._operator(' ', '[%s]' % degree)._sage_()
+                return ChainComplexMorphism(matrices, C, D)
         else:
             #Handle the integers and rationals separately
             if cls_str == "ZZ":
@@ -1526,9 +1573,22 @@ def _matrix_(self, R):
             sage: matrix(QQ, A)                          # optional - macaulay2, indirect doctest
             [1 2]
             [3 4]
+
+        TESTS:
+
+        Check that degenerate matrix dimensions are preserved (:trac:`28591`)::
+
+            sage: m = macaulay2('matrix {{},{}}')  # optional - macaulay2
+            sage: matrix(ZZ, m).dimensions()  # optional - macaulay2
+            (2, 0)
+            sage: matrix(ZZ, m.transpose()).dimensions()  # optional - macaulay2
+            (0, 2)
         """
         from sage.matrix.all import matrix
-        return matrix(R, self.entries()._sage_())
+        m = matrix(R, self.entries()._sage_())
+        if not m.nrows():
+            return matrix(R, 0, self.numcols()._sage_())
+        return m
 
 
 @instancedoc

src/sage/matrix/matrix1.pyx

@@ -447,10 +447,21 @@ cdef class Matrix(Matrix0):
             sage: m = macaulay2(matrix(R, [[1, 2], [3, 4]]))    # optional - macaulay2
             sage: m.ring()._operator('===', R).sage()           # optional - macaulay2
             True
+
+        Check that degenerate matrix dimensions are handled correctly
+        (:trac:`28591`)::
+
+            sage: macaulay2(matrix(QQ, 2, 0)).numrows()  # optional - macaulay2
+            2
+            sage: macaulay2(matrix(QQ, 0, 2)).numcols()  # optional - macaulay2
+            2
         """
         if macaulay2 is None:
             from sage.interfaces.macaulay2 import macaulay2 as m2_default
             macaulay2 = m2_default
+        if not self.nrows():
+            return (macaulay2(self.base_ring())
+                    .matrix('toList(%s:{})' % self.ncols()).transpose())
         entries = [list(row) for row in self]
         return macaulay2(self.base_ring()).matrix(entries)