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# Symbolic expression from a polynomial in a quotient ring

sage: R.<t> = PolynomialRing(ZZ)
sage: I = R.ideal([t^2-2])
sage: S.<z> = R.quotient_ring(I)
sage: eq_z = (z+1)^2-5; eq_z
2*z - 2
sage: SR(eq_z.lift()).variables()
(t,)
sage: SR(eq_z).variables()
()


I'd expect the last tuple to contain the variable z, what am I missing?

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## 1 answer

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Apparently the symbolic ring considers that SR(eq_z) is a constant.

Compare:

sage: p = (5*x^2-7*x+4)
sage: p.coefficients()
[[4, 0], [-7, 1], [5, 2]]


and

sage: R.<t> = PolynomialRing(ZZ)
sage: I = R.ideal([t^2-2])
sage: S.<z> = R.quotient_ring(I)
sage: eq_z = (z+1)^2-5; eq_z
2*z - 2
sage: a = SR(eq_z); a
2*z - 2
sage: a.coefficients()
[[2*z - 2, 0]]


So it's the constant 2*z-2 times x^0.

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Asked: 2014-12-16 13:58:43 -0600

Seen: 277 times

Last updated: Dec 16 '14