Revision history [back]
 | 1 | initial version |
Clarifying the question
If I understand correctly, the question is to extract
not the coefficient of the monomial z (which would be
zero in this case), but the coefficient of z
when this is seen as a polynomial in z
with coefficients in polynomials in x and y.
In other words the expected answer here is x*y + x.
Note that in Sage's symbolic ring, if a symbolic expression is a polynomial in one of the symbolic variables, then we can extract the coefficient as follows:
sage: x, y, z = SR.var('x, y, z')
sage: f = x*z + x*y*z + y + 1; f
x*y*z + x*z + y + 1
sage: f.coefficient(z, 1)
x*y + x
The question is how to do that when instead of a
symbolic expression in Sage's symbolic ring, our f
is an element in the quotient of a polynomial ring
in three variables modulo an ideal.
Setup
We reproduce the setup from the question with some variations, see notes below.
Define the field with three elements:
sage: K = GF(3); K
Finite Field of size 3
Define a polynomial ring in three variables:
sage: A.<X, Y, Z> = PolynomialRing(K); A
Multivariate Polynomial Ring in X, Y, Z
over Finite Field of size 3
Define an ideal in this polynomial ring:
sage: J = A.ideal(X^2 - 1, Y^2 - 1, Z^2 - 1)
Ideal (X^2 - 1, Y^2 - 1, Z^2 - 1)
of Multivariate Polynomial Ring in X, Y, Z
over Finite Field of size 3
Define the quotient ring:
sage: R.<x, y, z> = A.quotient(J); R
Quotient of
Multivariate Polynomial Ring in X, Y, Z
over Finite Field of size 3
by the ideal (X^2 - 1, Y^2 - 1, Z^2 - 1)
Define an element in the quotient ring:
sage: f = x*z + x*y*z + y + 1; f
x*y*z + x*z + y + 1
Coefficient of z using polynomial rings
Lift to the polynomial ring:
sage: F = f.lift(); F
X*Y*Z + X*Z + Y + 1
Define ring of polynomials in Z with coefficients
in polynomials in X and Y over K:
sage: B = K[('X', 'Y')]['Z']; B
Univariate Polynomial Ring in Z
over Multivariate Polynomial Ring in X, Y
over Finite Field of size 3
See F as that kind of polynomial:
sage: G = B(F); G
(X*Y + X)*Z + Y + 1
Coefficient of Z:
sage: D = G[1]; D
X*Y + X
Map back to polynomials in three variables:
sage: C = A(D); C
X*Y + X
Project down to the quotient ring:
sage: c = R(C); c
x*y + x
Note that this is an element in R;
we might prefer an element in the quotient
ring $K[X, Y] / (X^2 - 1, Y^2 - 1)$.
This is left as an exercise.
Using strings and the symbolic ring
We describe a different way to obtain the answer, using strings and the symbolic ring.
Lift to the polynomial ring:
sage: F = f.lift(); F
X*Y*Z + X*Z + Y + 1
Extract coefficient of (the symbolic variable) Z:
sage: C_SR = SR(str(F)).coefficient(SR('Z'), 1); C_SR
X*Y + X
Back to the polynomial ring:
sage: C = A(str(c_SR)); C
X*Y + X
Project to the quotient: sage: c = R(C); c x*y + x
Some notes
Integers modulo three vs the field with three elements
In Sage one gets different objects depending whether we construct the ring $\mathbb{Z}/3\mathbb{Z}$ or the field with three elements.
Here are four ways to construct the ring
sage: k = Zmod(3)
sage: k = Integers(3)
sage: k = IntegerModRing(3)
sage: k = ZZ.quo(3*ZZ)
All give the same result:
sage: k
Ring of integers modulo 3
Here are two ways to construct the field:
sage: K = FiniteField(3)
sage: K = GF(3)
Both give the same result:
sage: K
Finite Field of size 3
Although one can ask Sage whether k is a field,
and it will say yes, Sage does not consider k and K
as equal, and the arithmetic in them is not implemented
the same way.
sage: k.is_field()
True
sage: k == K
False
When computing with a field, it is recommended
to define it using FiniteField or GF, since
this will take better advantage of finite field
arithmetic.
Wishlist
While A can convert a string:
sage: A('X*Y*Z + X*Z + Y + 1')
X*Y*Z + X*Z + Y + 1
it is not the case of the quotient ring R:
sage: R('x*y*z + x*z + y + 1')
Traceback (most recent call last)
...
NameError: name 'x' is not defined
During handling of the above exception, another exception occurred:
Traceback (most recent call last)
...
TypeError: Could not find a mapping of the passed element to this ring.
It would be nice if one could get:
sage: R('x*y*z + x*z + y + 1')
x*y*z + x*z + y + 1
