1 | initial version |
Since 1
does not belong to the ideal, it is not possible to express 1
as a linear combination (with polynomial coefficients) of the generators of the ideal (which is what lift
tries to do).
So lift
correctly gives an error; it even provides the reason.
2 | No.2 Revision |
Since 1
does not belong to the ideal, it is not possible to express 1
as a linear combination (with polynomial coefficients) of the generators of the ideal (which is what lift
tries to do).
So lift
correctly gives an error; it even provides the reason.
Edit (after the generators were changed): you should work over a field with exact arithmetic, such as the smallest field that contains the coefficients of your polynomials; in this case you can work over QQ
:
sage: A.<x, y> = PolynomialRing(QQ, 2, order='degrevlex')
sage: I = A.ideal([x + y, x + y + 1])
sage: A(1).lift(I)
[-1, 1]
This result implies the result over $\mathbb{C}$.