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Often I want to group terms in different ways. Say for example, I wish to view my equation in terms of some variables $u, v, w$.

R.<u, v, w> = SR[]; var("α a13 a21 a23 a1 a3 a2 a4 a6")
x = α^2*u + a13*w;   y = a21*u + α^3*v + a23*w;  z = w
e = y^2*z + a1*x*y*z + a3*y*z^2 - (x^3 + a2*x^2*z + a4*x*z^2 + a6*z^3)
e

which gives me an expression expressed in terms of the generators $u, v, w$ as desired:

$$(-α^6)u^3 + (-3a_{13}α^4 - a_2α^4 + a_1a_{21}α^2 + a_{21}^2)u^2w + (a_1α^5 + 2a_{21}α^3)uvw + α^6v^2w + (-3a_{13}^2α^2 - 2a_{13}a_2α^2 + a_1a_{23}α^2 + a_1a_{13}a_{21} - a_4α^2 + 2a_{21}a_{23} + a_{21}a_3)uw^2 + (a_1a_{13}α^3 + 2a_{23}α^3 + a_3α^3)vw^2 + (-a_{13}^3 - a_{13}^2a_2 + a_1a_{13}a_{23} + a_{23}^2 + a_{23}a_3 - a_{13}a_4 - a_6)w^3 $$

However $\alpha^6 = 1$, so I want some way of expressing this, so I construct a quotient extension ring (please let me know if there's a better way):

R.<u, v, w> = SR[]; var("a13 a21 a23 a1 a3 a2 a4 a6")
S.<p> = R[]
T.<α> = S.quotient(p^6 - 1)
x = α^2*u + a13*w;   y = a21*u + α^3*v + a23*w;  z = w
e = y^2*z + a1*x*y*z + a3*y*z^2 - (x^3 + a2*x^2*z + a4*x*z^2 + a6*z^3)
e

$$a_1uvwα^5 + ((-3a_{13} - a_2)u^2w)α^4 + (2a_{21}uvw + (a_1a_{13} + 2a_{23} + a_3)vw^2)α^3 + (a_1a_{21}u^2w + (-3a_{13}^2 - 2a_{13}a_2 + a_1a_{23} - a_4)uw^2)α^2 - u^3 + a_{21}^2u^2w + v^2w + (a_1a_{13}a_{21} + 2a_{21}a_{23} + a_{21}a_3)uw^2 + (-a_{13}^3 - a_{13}^2a_2 + a_1a_{13}a_{23} + a_{23}^2 + a_{23}a_3 - a_{13}a_4 - a_6)w^3$$ which now consists of expressions in $u, v, w$ in $\alpha$, whereas I just want the expression in $u, v, w$.

I now try coercing $\alpha$ to a generic variable:

sage: var("q")
sage: e(α=q)
TypeError: 'PolynomialQuotientRing_generic_with_category.element_class' object is not callable
sage: e.lift()(p=S.coerce(q))

but the resulting expression still consists of expressions in $a_i$ over $u, v, w$ over $q$. Maybe I have to now convert it to a purely symbolic expression, and then convert it back to $R.<u, v,="" w="">$?

var("u1 v1 w1")
e.lift()(p=S.coerce(q), u=u1, v=v1, w=w1)(u1=u, v1=v, w1=w)

Unfortunately this does not work for me. I just get an expression in u1, v1, w1... Any ideas?

Often I want to group terms in different ways. Say for example, I wish to view my equation in terms of some variables $u, v, w$.

R.<u, v, w> = SR[]; var("α a13 a21 a23 a1 a3 a2 a4 a6")
x = α^2*u + a13*w;   y = a21*u + α^3*v + a23*w;  z = w
e = y^2*z + a1*x*y*z + a3*y*z^2 - (x^3 + a2*x^2*z + a4*x*z^2 + a6*z^3)
e

which gives me an expression expressed in terms of the generators $u, v, w$ as desired:

$$(-α^6)u^3 + (-3a_{13}α^4 - a_2α^4 + a_1a_{21}α^2 + a_{21}^2)u^2w + (a_1α^5 + 2a_{21}α^3)uvw + α^6v^2w + (-3a_{13}^2α^2 - 2a_{13}a_2α^2 + a_1a_{23}α^2 + a_1a_{13}a_{21} - a_4α^2 + 2a_{21}a_{23} + a_{21}a_3)uw^2 + (a_1a_{13}α^3 + 2a_{23}α^3 + a_3α^3)vw^2 + (-a_{13}^3 - a_{13}^2a_2 + a_1a_{13}a_{23} + a_{23}^2 + a_{23}a_3 - a_{13}a_4 - a_6)w^3 $$

However $\alpha^6 = 1$, so I want some way of expressing this, so I construct a quotient extension ring (please let me know if there's a better way):

R.<u, v, w> = SR[]; var("a13 a21 a23 a1 a3 a2 a4 a6")
S.<p> = R[]
T.<α> = S.quotient(p^6 - 1)
x = α^2*u + a13*w;   y = a21*u + α^3*v + a23*w;  z = w
e = y^2*z + a1*x*y*z + a3*y*z^2 - (x^3 + a2*x^2*z + a4*x*z^2 + a6*z^3)
e

$$a_1uvwα^5 + ((-3a_{13} - a_2)u^2w)α^4 + (2a_{21}uvw + (a_1a_{13} + 2a_{23} + a_3)vw^2)α^3 + (a_1a_{21}u^2w + (-3a_{13}^2 - 2a_{13}a_2 + a_1a_{23} - a_4)uw^2)α^2 - u^3 + a_{21}^2u^2w + v^2w + (a_1a_{13}a_{21} + 2a_{21}a_{23} + a_{21}a_3)uw^2 + (-a_{13}^3 - a_{13}^2a_2 + a_1a_{13}a_{23} + a_{23}^2 + a_{23}a_3 - a_{13}a_4 - a_6)w^3$$ which now consists of expressions in $u, v, w$ in $\alpha$, whereas I just want the expression in $u, v, w$.

I now try coercing $\alpha$ to a generic variable:

sage: var("q")
sage: e(α=q)
TypeError: 'PolynomialQuotientRing_generic_with_category.element_class' object is not callable
sage: e.lift()(p=S.coerce(q))

but the resulting expression still consists of expressions in $a_i$ over $u, v, w$ over $q$. Maybe I have to now convert it to a purely symbolic expression, and then convert it back to $R.<u, v,="" w="">$?R.<u, v, w>?

var("u1 v1 w1")
e.lift()(p=S.coerce(q), u=u1, v=v1, w=w1)(u1=u, v1=v, w1=w)

Unfortunately this does not work for me. I just get an expression in u1, v1, w1... Any ideas?

Often I want to group terms in different ways. Say for example, I wish to view my equation in terms of some variables $u, v, w$.

R.<u, v, w> = SR[]; var("α a13 a21 a23 a1 a3 a2 a4 a6")
x = α^2*u + a13*w;   y = a21*u + α^3*v + a23*w;  z = w
e = y^2*z + a1*x*y*z + a3*y*z^2 - (x^3 + a2*x^2*z + a4*x*z^2 + a6*z^3)
e

which gives me an expression expressed in terms of the generators $u, v, w$ as desired:

$$(-α^6)u^3 + (-3a_{13}α^4 - a_2α^4 + a_1a_{21}α^2 + a_{21}^2)u^2w + (a_1α^5 + 2a_{21}α^3)uvw + α^6v^2w + (-3a_{13}^2α^2 - 2a_{13}a_2α^2 + a_1a_{23}α^2 + a_1a_{13}a_{21} - a_4α^2 + 2a_{21}a_{23} + a_{21}a_3)uw^2 + (a_1a_{13}α^3 + 2a_{23}α^3 + a_3α^3)vw^2 + (-a_{13}^3 - a_{13}^2a_2 + a_1a_{13}a_{23} + a_{23}^2 + a_{23}a_3 - a_{13}a_4 - a_6)w^3 $$

However $\alpha^6 = 1$, so I want some way of expressing this, so I construct a quotient extension ring (please let me know if there's a better way):

R.<u, v, w> = SR[]; var("a13 a21 a23 a1 a3 a2 a4 a6")
S.<p> = R[]
T.<α> = S.quotient(p^6 - 1)
x = α^2*u + a13*w;   y = a21*u + α^3*v + a23*w;  z = w
e = y^2*z + a1*x*y*z + a3*y*z^2 - (x^3 + a2*x^2*z + a4*x*z^2 + a6*z^3)
e

$$a_1uvwα^5 + ((-3a_{13} - a_2)u^2w)α^4 + (2a_{21}uvw + (a_1a_{13} + 2a_{23} + a_3)vw^2)α^3 + (a_1a_{21}u^2w + (-3a_{13}^2 - 2a_{13}a_2 + a_1a_{23} - a_4)uw^2)α^2 - u^3 + a_{21}^2u^2w + v^2w + (a_1a_{13}a_{21} + 2a_{21}a_{23} + a_{21}a_3)uw^2 + (-a_{13}^3 - a_{13}^2a_2 + a_1a_{13}a_{23} + a_{23}^2 + a_{23}a_3 - a_{13}a_4 - a_6)w^3$$ which now consists of expressions in $u, v, w$ in $\alpha$, whereas I just want the expression in $u, v, w$.

I now try coercing $\alpha$ to a generic variable:

sage: var("q")
sage: e(α=q)
TypeError: 'PolynomialQuotientRing_generic_with_category.element_class' object is not callable
sage: e.lift()(p=S.coerce(q))

but the resulting expression still consists of expressions in $a_i$ over $u, v, w$ over $q$. Maybe I have to now convert it to a purely symbolic expression, and then convert it back to R.<u, v, w>?

var("u1 v1 w1")
e.lift()(p=S.coerce(q), u=u1, v=v1, w=w1)(u1=u, v1=v, w1=w)

Unfortunately this does not work for me. I just get an expression in u1, v1, w1... Any ideas?

sage: var("q")
sage: sum(a*q^i for i, a in enumerate(e.lift().coefficients(sparse=False)))