> > - - multivariate polynomials:
> > given an ideal I = (f_1,...,f_r) and some g in I, find s_1,...,s_r such
> > that g = s_1 f_1 + ... + s_r f_r (this doesn't seem to be available
> > even for Groebner basis)
>
> You're right that this doesn't seem to be available, which is a strange
> oversight. It *must* be easily available within Singular though, so it's
> probably a few lines of code to add it to SAGE.
See attachment.
--
name: Martin Albrecht
_pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99
_www: http://www.informatik.uni-bremen.de/~malb
_jab: [EMAIL PROTECTED]
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# HG changeset patch
# User 'Martin Albrecht <[EMAIL PROTECTED]>'
# Date 1170879646 28800
# Node ID 289e32a37192e033c54e2edcc4b78b81a1f8bec2
# Parent b5f4f6a1d90dd14f32e2641a1a87d86b81e625e6
added lift method.
diff -r b5f4f6a1d90d -r 289e32a37192 sage/rings/multi_polynomial_element.py
--- a/sage/rings/multi_polynomial_element.py Tue Feb 06 21:34:28 2007 -0800
+++ b/sage/rings/multi_polynomial_element.py Wed Feb 07 12:20:46 2007 -0800
@@ -49,6 +49,8 @@ from sage.structure.factorization import
from sage.structure.factorization import Factorization
from sage.rings.polynomial_singular_interface import Polynomial_singular_repr
+
+from sage.structure.sequence import Sequence
import multi_polynomial_ring
import polynomial_ring
@@ -1023,7 +1025,33 @@ class MPolynomial_polydict(Polynomial_si
F = Factorization(v)
F.sort()
return F
-
+
+ def lift(self,I):
+ """
+ given an ideal I = (f_1,...,f_r) and some g (== self) in I,
+ find s_1,...,s_r such that g = s_1 f_1 + ... + s_r f_r
+
+ ALGORITHM: Use Singular.
+
+ EXAMPLE:
+ sage: A.<x,y> = PolynomialRing(QQ,2,order='degrevlex')
+ sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
+ sage: f = x*y^13 + y^12
+ sage: M = f.lift(I)
+ sage: M
+ [y^4 + x*y^5 + x^2*y^3 + x^3*y^4 + x^4*y^2 + x^5*y^3 + x^6*y + x^7*y^2 + x^8, y^7]
+ sage: sum( map( mul , zip( M, I.gens() ) ) ) == f
+ True
+ """
+ fs = self._singular_()
+ Is = I._singular_()
+ P = I.ring()
+ try:
+ M = Is.lift(fs)._sage_(P)
+ except TypeError:
+ raise ArithmeticError, "f is not in I"
+ return Sequence(M.list(), P, check=False, immutable=True)
+
def gcd(self, f):
"""
Compute the greatest common divisor of this polynomial and f.
