Expanded Reflexive strategy of Matrix // Matrix

M2/Macaulay2/m2/matrix2.m2

@@ -396,6 +396,7 @@ Matrix \\ Matrix := Matrix => (g, f) -> quotient'(f, g)
 quotient(Matrix, Matrix) := Matrix => opts -> (f, g) -> (
     -- given f: A-->C and g: B-->C, then find (f//g): A-->B such that g o (f//g) + r = f
     if target f != target g then error "quotient: expected maps with the same target";
+    if f == 0 then return map(source g, source f, 0);
     c := runHooks((quotient, Matrix, Matrix), (opts, f, g), Strategy => opts.Strategy);
     if c =!= null then c else error "quotient: no method implemented for this type of input")
 
@@ -408,11 +409,19 @@ addHook((quotient, Matrix, Matrix), Strategy => Default,
 	    MinimalGenerators => opts.MinimalGenerators };
 	map(source g, source f, homomorphism(homomorphism'(f, opts) // Hom(source f, g, opts)))))
 
--- FIXME: this is still causing unreasonable slow downs, e.g. for (large m) // (scalar)
-addHook((quotient, Matrix, Matrix), Strategy => "Reflexive", (opts, f, g) -> if f == 0 or isFreeModule source f then (
-     L := source f;	     -- result may not be well-defined if L is not free
+addHook((quotient, Matrix, Matrix), Strategy => "Reflexive", (opts, f, g) -> (
+     L := source f;
      M := target f;
      N := source g;
+     -- TODO: should this be a separate strategy?
+     if not isFreeModule L then return (
+	 -- result may not be well-defined if L is not free,
+	 -- unless the composition h * syz p is zero
+	 p := coverMap L;
+	 h := quotient(f * p, g, Strategy => "Reflexive");
+	 -- TODO: does h * gens ker p != 0 suffice?
+	 if h * inducedMap(source p, kernel p) == 0 then map(N, L, h));
+     --
      if M.?generators then (
 	  M = cokernel presentation M;	    -- this doesn't change the cover
 	  );

M2/Macaulay2/packages/Macaulay2Doc/functions/quotient-remainder-doc.m2

@@ -152,8 +152,8 @@ Node
       reduction of @TT "f"@ modulo a Gröbner basis for the image of @TT "dual g"@.
       See @TO quotient@ for the dual notion.
 
-      If the remainder @TT "f - h*g"@ is zero, then the quotient @TT "g\\f"@
-      satisfies the equation @TT "f === (g\\f) * g"@. Otherwise, the equation
+      If the remainder @TT "f - h*g"@ is zero, then the quotient @TT "g\\\\f"@
+      satisfies the equation @TT "f === (g\\\\f) * g"@. Otherwise, the equation
       @TT "h * g + r === f"@ will hold, where @TT "r"@ is the map provided by
       @TO remainder'@.
     Example

M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2

@@ -1870,7 +1870,12 @@ document { Key => "loadedFiles",
      PARA { "After each source file is successfully loaded, the full path to the file is stored in the hash table ", TO "loadedFiles", ".  It is stored as the
 	  value, with the corresponding key being a small integer, consecutively assigned, starting at 0."
 	  },
-     EXAMPLE "peek loadedFiles"}
+     EXAMPLE lines ///
+	loadedFiles#0
+        #loadedFiles
+     ///,
+     --SeeAlso => { "filesLoaded" },
+     }
 
 document { Key => "homeDirectory",
      Headline => "the home directory of the user",

M2/Macaulay2/tests/normal/quotient.m2

@@ -40,3 +40,117 @@ assert(f \\ h == g)
 assert((f\\h) * f == h)
 assert(g // h == 0) -- does it always happen that this is zero if lift can't occur?  Probably not
 assert(h \\ f == 0)
+
+
+-- tests which show that the "Reflexive" strategy doesn't always work
+S = QQ[x_0,x_1,x_2] --/ sum(3, i -> x_i^3)
+a = x_0 + x_1
+b = x_0^2-x_0*x_1+x_1^2
+A = matrix {{x_2, a}, {b, -x_2^2}}
+B = matrix {{x_2^2, a}, {b, -x_2}}
+
+f = map(S^2 / a, S^2 / a, A * B)
+f' = map(S^2 / a, S^2, A * B)
+g = map(S^2 / a, S^2, A)
+h = map(S^2 / a, S^2 / a, B)
+assert all({f', f, g, h}, isWellDefined)
+
+-- f does not factor through g without a remainder
+assert not isSubset(image homomorphism' f, image Hom(source f, g))
+
+-- here is the example
+q = f // g
+assert(isWellDefined q and q == 0)
+
+q = quotient(f, g, Strategy => Default)
+assert(isWellDefined q and q == 0)
+
+-- the Reflexive strategy is not applicable here
+assert try ( quotient(f, g, Strategy => "Reflexive"); false ) else true
+
+-- but f' does (left) factor through g
+assert isSubset(image homomorphism' f, image Hom(g, target f))
+
+q = g \\ f'
+assert isWellDefined q
+assert(f' == q * g)
+assert(h == q)
+
+q = quotient'(f', g, Strategy => Default)
+assert isWellDefined q
+assert(f' == q * g)
+assert(h == q)
+
+assert try ( quotient'(f', g, Strategy => "Reflexive"); false ) else true
+
+
+-- this is an example where the improved "Reflexive" strategy works
+f = homomorphism random Hom(image(A | B), S^2)
+g = homomorphism random Hom(image(B | A), S^2)
+h = homomorphism random Hom(image(A | B), image(B | A))
+f = g * h
+assert all({f, g}, isWellDefined)
+
+-- f factors through g without a remainder
+assert isSubset(image homomorphism' f, image Hom(source f, g))
+
+-- here is the example
+q = f // g
+assert(isWellDefined q and f == g * q)
+
+q = quotient(f, g, Strategy => Default)
+assert(isWellDefined q and f == g * q)
+
+
+-- this is an example where the improved "Reflexive" strategy works
+R = ZZ/2[x,y,z,w]
+g = matrix {{x, x+y+z}, {x+y+z, z}}
+f = g * matrix {{y^2}, {z^2}}
+f = inducedMap(target f / (x+y+z), source f / (x+y+z), f)
+g = inducedMap(target g / (x+y+z), source g, g)
+isWellDefined f
+isWellDefined g
+
+-- f does not factor through g without a remainder
+assert not isSubset(image homomorphism' f, image Hom(source f, g))
+
+q = quotient(f, g, Strategy => Default)
+assert(isWellDefined q and q == 0)
+
+assert try ( quotient(f, g, Strategy => "Reflexive"); false ) else true
+
+----
+clearAll
+B = QQ[a..d]
+f = prune map(B^1, image(map(B^{{-2}, 3:{-3}}, B^{{-3}, 3:{-4}, {-3}, 3:{-4}, {-3}, 3:{-4}, {-3}, 3:{-4}}, {{a, 0, 0, 0, b, 0, 0, 0, c, 0, 0, 0, d, 0, 0, 0}, {0, a, 0, 0, 0, b, 0, 0, 0, c, 0, 0, 0, d, 0, 0}, {0, 0, a, 0, 0, 0, b, 0, 0, 0, c, 0, 0, 0, d, 0}, {0, 0, 0, a, 0, 0, 0, b, 0, 0, 0, c, 0, 0, 0, d}})),{{a*b*c-a^2*d, a*b^3-a^3*c, a^2*c^2-a*b^2*d, a*c^3-a*b*d^2, b^2*c-a*b*d, b^4-a^2*b*c, a*b*c^2-b^3*d, b*c^3-b^2*d^2, b*c^2-a*c*d, b^3*c-a^2*c^2, a*c^3-b^2*c*d, c^4-b*c*d^2, b*c*d-a*d^2, b^3*d-a^2*c*d, a*c^2*d-b^2*d^2, c^3*d-b*d^3}})
+g = prune map(B^1, image(map(B^1, B^{4:{-1}}, {{a, b, c, d}})), {{a, b, c, d}})
+assert isSubset(image homomorphism' f, image Hom(source f, g))
+elapsedTime assert(f == g * quotient(f, g, Strategy => Default))
+elapsedTime assert(f == g * quotient(f, g, Strategy => "Reflexive"))
+
+----
+S = QQ[x]
+f = random(S^1/x, S^1/x^3)
+g = random(S^1/x, S^1/x^2)
+elapsedTime assert(f == g * quotient(f, g, Strategy => Default))
+elapsedTime assert(f == g * quotient(f, g, Strategy => "Reflexive"))
+
+----
+S = QQ[x,y];
+M = coker matrix {{x,y},{-y,x}};
+g = random(M, M)
+f = g * random(M, M)
+q = quotient(f, g, Strategy => "Reflexive")
+assert(isWellDefined q and f == g * q)
+
+----
+S = QQ[x,y,z]
+M = S^{-2}
+N = S^{-3}
+P = truncate(3, S^1)
+h = homomorphism random(3, Hom(P, N))
+g = homomorphism random(0, Hom(N, M))
+f = g * h
+assert all({f, g, h}, isWellDefined)
+q = quotient(f, g, Strategy => "Reflexive")
+assert(isWellDefined q and f == g * q)

M2/cmake/build-libraries.cmake

@@ -511,7 +511,7 @@ ExternalProject_Add(build-msolve
   TEST_EXCLUDE_FROM_MAIN ON
   STEP_TARGETS      install test
   )
-_ADD_COMPONENT_DEPENDENCY(programs msolve "gmp;mpfr;flint" MSOLVE_FOUND)
+_ADD_COMPONENT_DEPENDENCY(programs msolve "gmp;mpfr;flint" MSOLVE)
 
 
 # https://numpi.dm.unipi.it/software/mpsolve