# How to base change from a PolynomialRing to that Ring with one variable evaluated, i.e., from Q[x,y] to Q[x,y]/(x=0) = Q[y]?

I am trying to base change a Laurent series ring element from its base ring, Q[u1, u2, u3], to a quotient of its base ring, Q[u2, u3], but I am quite confused in forming this quotient. My setting is this:

S.<u1,u2,u3> = QQ[]
L.<z> = LaurentSeriesRing(S);
f = -4*z - 4/5*u1*z^5 + (-4/9*u1^2 - 8/9*u2)*z^9 + (-4/13*u1^3 - 24/13*u1*u2 - 12/13*u3)*z^13 + (-4/17*u1^4 - 48/17*u1^2*u2 - 24/17*u2^2 - 48/17*u1*u3 + 16/17*u1 + 16/17*u2 + 16/17*u3 + 16/17)*z^17 + O(z^20)


In other words, I wish to set u1 = 0, and look at f over that ring. I tried the following two things, which spit out f unchanged.

f.change_ring(S.quo(u1))

R = S.quotient(u1)
f.change_ring(R)


I also tried the following which gives an attribute error:

f.reduce(u1)


I am completely stuck and would deeply appreciate any help. I would also like to eventually set u2 = 0 and look at f over that ring, which I mention at the off chance that this changes the answer at all.

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This is one way:

sage: LL = L.change_ring(S.quo(u3))
sage: LL
Laurent Series Ring in z over Quotient of Multivariate Polynomial Ring in u1, u2, u3 over Rational Field by the ideal (u3)
sage: LL(f)
-4*z - 4/5*u1bar*z^5 + (-4/9*u1bar^2 - 8/9*u2bar)*z^9 + (-4/13*u1bar^3 - 24/13*u1bar*u2bar)*z^13 + (-4/17*u1bar^4 - 48/17*u1bar^2*u2bar - 24/17*u2bar^2 + 16/17*u1bar + 16/17*u2bar + 16/17)*z^17 + O(z^20)

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