Changeset 5190:daab147382bc

Timestamp:

06/25/07 14:54:34 (6 years ago)

Author:

'Martin Albrecht <malb@…

Branch:

default

Parents:

5188:a80ccb008dea (diff), 5189:df988cf43484 (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.

Message:

merge

Files:

• sage/rings/polynomial/multi_polynomial_ideal.py (modified) (3 diffs)
• sage/rings/polynomial/multi_polynomial_ideal.py (modified) (8 diffs)

sage/rings/polynomial/multi_polynomial_ideal.py

@@ -127 +127 @@
 from sage.misc.misc import prod
 import sage.rings.integer_ring
-
+import sage.rings.polynomial.toy_buchberger as toy_buchberger
 
 def is_MPolynomialIdeal(x):
@@ -954 +954 @@
                          * 'singular:stdfglm' - Singular's stdfglm command
                          * 'singular:slimgb' - Singular's slimgb command
+                         * 'toy:buchberger' - SAGE's toy/educational buchberger without strategy
+                         * 'toy:buchberger2' - SAGE's toy/educational buchberger with strategy
                          * 'macaulay2:gb' (if available) - Macaulay2's gb command
                          * 'magma:GroebnerBasis' (if available) - MAGMA's Groebnerbasis command
 
-        ALGORITHM: Uses Singular, MAGMA, or Macaulay2 (if available)
+        EXAMPLES:
+            Consider Katsura-3 over QQ with term ordering 'degrevlex'
+
+            sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis()
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]
+
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('singular:groebner')
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('singular:std')
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('singular:stdhilb')
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 5*b^2 - b - 3*c^2 + c, a + 2*b + 2*c - 1]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('singular:stdfglm')
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('singular:slimgb')
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('toy:buchberger')
+            [-30*c^4 + 100/7*c^3 - 5/14*c^2 - 5/14*c, -2*b^2 - b*c + 1/2*b, a + 2*b + 2*c - 1, 
+             7/125*b + 42/25*c^3 - 79/125*c^2 + 3/125*c, a^2 - a + 2*b^2 + 2*c^2, 
+             -5*b*c + 1/2*b - 6*c^2 + 2*c, 2*a*b + 2*b*c - b]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('toy:buchberger2')
+            [30*c^4 - 100/7*c^3 + 5/14*c^2 + 5/14*c, a + 2*b + 2*c - 1, 
+             7/125*b + 42/25*c^3 - 79/125*c^2 + 3/125*c, a^2 - a + 2*b^2 + 2*c^2]
+
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('macaulay2:gb') # optional requires Macaulay2
+            [84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]
+            
+            sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
+            sage: I.groebner_basis('magma:GroebnerBasis') # optional requires MAGMA
+            [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
+
+            If Macaulay2 is installed, Groebner bases over ZZ can be computed.
+
+            sage: P.<a,b,c> = PolynomialRing(ZZ,3)
+            sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
+            sage: I.groebner_basis() #optional requires Macaulay2 
+            [a + 2*b + 2*c - 1, 10*b*c + 12*c^2 - b - 4*c, 2*b^2 - 4*b*c - 6*c^2 + 2*c, 
+             42*c^3 + b^2 + 2*b*c - 14*c^2 + b, 2*b*c^2 - 6*c^3 + b^2 + 5*b*c + 8*c^2 - b - 2*c,
+             b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2]
+
+            SAGE also supports local orderings (via SINGULAR):
+
+            sage: P.<x,y,z> = PolynomialRing(QQ,3,order='negdegrevlex')
+            sage: I = P * (  x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 +y ^5 )
+            sage: I.groebner_basis()
+            [2*x^2 + y^3, x*y*z + z^5, y^5 + 3*z^5, y^4*z - 2*x*z^5, z^6]
+
+        ALGORITHM: Uses Singular, MAGMA (if available), Macaulay2 (if
+        available), or toy implementation.
 
         """
@@ -971 +1038 @@
         elif algorithm == 'magma:GroebnerBasis':
             return self._magma_groebner_basis()
+        elif algorithm == 'toy:buchberger':
+            return toy_buchberger.buchberger(self)
+        elif algorithm == 'toy:buchberger2':
+            return toy_buchberger.buchberger_improved(self)
         else:
             raise TypeError, "algorithm '%s' unknown"%algorithm

sage/rings/polynomial/multi_polynomial_ideal.py

@@ -1 +1 @@
 """
 Ideals in multivariate polynomial rings.
+
+Most functionality of multivariate polynomial ideals in SAGE is
+provided through SINGULAR.
 
 AUTHOR:
@@ -6 +9 @@
     -- Kiran S. Kedlaya (2006-02-12): added Macaulay2 analogues of
               some Singular features
-    -- Martin Albrecht (2006-08-28): reorganized class hierarchy
+    -- Martin Albrecht
 
 EXAMPLES:
@@ -122 +125 @@
 from sage.structure.sequence import Sequence
 from sage.misc.sage_eval import sage_eval
+from sage.misc.misc import prod
 import sage.rings.integer_ring
 import sage.rings.polynomial.toy_buchberger as toy_buchberger
@@ -224 +228 @@
     def _contains_(self, f):
         """
+        Returns True if f is in the ideal self.
+        
         EXAMPLES:
             sage: R, (x,y) = PolynomialRing(QQ, 2, 'xy').objgens()
@@ -233 +239 @@
             sage: x^3 + 2*y in I
             True
+
+        NOTE: Requires computation of a Groebner basis, which is a
+        very expensive operation.
         """
 
@@ -353 +362 @@
         """
         The dimension of the ring modulo this ideal.
+
+        NOTE: Requires computation of a Groebner basis, which is a
+        very expensive operation.
         """
         try:
@@ -592 +604 @@
             sage: (y^2 - x)^2
             y^4 - 2*x*y^2 + x^2
+
+        NOTE: Requires computation of a Groebner basis, which is a
+        very expensive operation.
         """
         if self.base_ring() == sage.rings.integer_ring.ZZ:
@@ -769 +784 @@
         else:
             raise TypeError, "Cannot convert basis with given algorithm"
-            
+
+    def elimination_ideal(self, variables):
+        """
+        Returns the elimination ideal of self with respect to the
+        variables given in 'variables'.
+
+        INPUT:
+            variables -- a list or tuple of variables in self.ring()
+
+        EXAMPLE:
+            sage: R.<x,y,t,s,z> = PolynomialRing(QQ,5)
+            sage: I = R * [x-t,y-t^2,z-t^3,s-x+y^3]
+            sage: I.elimination_ideal([t,s])
+            Ideal (y^2 - x*z, x*y - z, x^2 - y) of Polynomial Ring in x, y, t, s, z over Rational Field
+
+        ALGORITHM: Uses SINGULAR
+
+        NOTE: Requires computation of a Groebner basis, which is a
+        very expensive operation.
+        """
+        if not isinstance(variables, (list,tuple)):
+            variables = (variables,)
+        
+        try:
+            Is = self.__singular_groebner_basis
+        except AttributeError:
+            Is = self._singular_()
+
+        R = self.ring()
+        return MPolynomialIdeal(R, [f.sage_poly(R) for f in Is.eliminate( prod(variables) ) ] )
 
 class MPolynomialIdeal_macaulay2_repr: