X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fpolynomial.py;h=8b493711924d705e17eb612efd56e390ea98e88e;hb=892a138853b8152e35605deedbfc8160309c2bc6;hp=7414bdf16b33e8178c746a3edf5b6fa4dd3f985b;hpb=4e4efc9eff5c77a2ca19002b1dfa45598e974c54;p=sage.d.git

diff --git a/mjo/polynomial.py b/mjo/polynomial.py
index 7414bdf..8b49371 100644
--- a/mjo/polynomial.py
+++ b/mjo/polynomial.py
@@ -75,6 +75,61 @@ def multidiv(f, gs):
         ....:          for g in gs )
         True
 
+    A solution ``g`` to Exercise 6 in Section 2.3 of Cox, Little, and
+    O'Shea that lives in the ideal generated by ``f1`` and ``f2`` but
+    which has nonzero remainder after division::
+
+        sage: R = PolynomialRing(QQ, 'x,y', order='deglex')
+        sage: x,y = R.gens()
+        sage: f1 = 2*x*y^2 - x
+        sage: f2 = 3*x^2*y - y - 1
+        sage: I = R.ideal(f1,f2)
+        sage: g = 2*y*f2
+        sage: g in I
+        True
+        sage: (qs,r) = multidiv(g,[f1,f2])
+        sage: r.is_zero()
+        False
+
+    Two solutions ``g`` to Exercise 7 in Section 2.3 of Cox, Little, and
+    O'Shea that live in the ideal generated by ``f1``, ``f2``, and ``f3``
+    but which have nonzero remainders after division::
+
+        sage: R = PolynomialRing(QQ, 'x,y,z', order='deglex')
+        sage: x,y,z = R.gens()
+        sage: f1 = x^4*y^2 - z
+        sage: f2 = x^3*y^3 - 1
+        sage: f3 = x^2*y^4 - 2*z
+        sage: I = R.ideal(f1,f2,f3)
+        sage: g = x^2*f3
+        sage: g in I
+        True
+        sage: (qs, r) = multidiv(g, [f1,f2,f3])
+        sage: r.is_zero()
+        False
+        sage: g = x*f2
+        sage: g in I
+        True
+        sage: (qs, r) = multidiv(g, [f1,f2,f3])
+        sage: r.is_zero()
+        False
+
+    Example 4 in Section 2.3 of Cox, Little, and O'Shea. This is the
+    same as Example 2, except with the order of ``gs`` reversed::
+
+        sage: R = PolynomialRing(QQ, 'x,y', order='lex')
+        sage: x,y = R.gens()
+        sage: f = x^2*y + x*y^2 + y^2
+        sage: gs = [ y^2 - 1, x*y - 1 ]
+        sage: (qs, r) = multidiv(f, gs)
+        sage: (qs, r)
+        ([x + 1, x], 2*x + 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:
 
     If we get a zero remainder, then the numerator should belong to