mjo/polynomial.py: improve tests for new multidiv function.

diff --git a/mjo/polynomial.py b/mjo/polynomial.py
index 3e90ba5bf3cf3862360ed5d5e5e3ccdcd9c17647..c3fbf56a475f7f1ee75fecc4c6d63fae16196f68 100644 (file)
--- a/mjo/polynomial.py
+++ b/mjo/polynomial.py
@@ -51,8 +51,14 @@ def multidiv(f, gs):
         sage: x,y = R.gens()
         sage: f = x*y^2 + 1
         sage: gs = [ x*y + 1, y + 1 ]
-        sage: multidiv(f, gs)
+        sage: (qs, r) = multidiv(f, gs)
+        sage: (qs, r)
         ([y, -1], 2)
+        sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+        True
+        sage: not any( g.lt().divides(m) for m in r.monomials()
+        ....:          for g in gs )
+        True
 
     Example 2 in Section 2.3 of Cox, Little, and O'Shea::
 
@@ -60,12 +66,46 @@ def multidiv(f, gs):
         sage: x,y = R.gens()
         sage: f = x^2*y + x*y^2 + y^2
         sage: gs = [ x*y - 1, y^2 - 1 ]
-        sage: multidiv(f, gs)
+        sage: (qs, r) = multidiv(f, gs)
+        sage: (qs, r)
         ([x + y, 1], x + y + 1)
+        sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+        True
+        sage: not any( g.lt().divides(m) for m in r.monomials()
+        ....:          for g in gs )
+        True
 
     TESTS:
 
-    Derp.
+    If we get a zero remainder, then the numerator should belong to
+    the ideal generated by the denominators::
+
+        sage: set_random_seed()
+        sage: R = PolynomialRing(QQ, 'x,y,z')
+        sage: x,y,z = R.gens()
+        sage: s = ZZ.random_element(1,5).abs()
+        sage: gs = [ R.random_element() for idx in range(s) ]
+        sage: f = R.random_element(ZZ.random_element(10).abs())
+        sage: (qs, r) = multidiv(f,gs)
+        sage: r != 0 or f in R.ideal(gs)
+        True
+
+    The numerator is always the sum of the remainder and the quotients
+    times the denominators, and the remainder's monomials aren't divisible
+    by the leading term of any denominator::
+
+        sage: set_random_seed()
+        sage: R = PolynomialRing(QQ, 'x,y,z')
+        sage: x,y,z = R.gens()
+        sage: s = ZZ.random_element(1,5).abs()
+        sage: gs = [ R.random_element() for idx in range(s) ]
+        sage: f = R.random_element(ZZ.random_element(10).abs())
+        sage: (qs, r) = multidiv(f,gs)
+        sage: r + sum( qs[i]*gs[i] for i in range(len(gs)) ) == f
+        True
+        sage: r == 0 or (not any( g.lt().divides(m) for m in r.monomials()
+        ....:                     for g in gs ))
+        True
 
     """
     R = f.parent()