Inconsistentency in parent of specialization of a polynomial?

asked 2020-12-15 04:44:22 -0600

mvk gravatar image

updated 2020-12-15 04:59:32 -0600

I have a family of polynomials and I want to consider special members of this family. In other words I'm considering polynomials in a ring $R = K[x]$ where $K = \mathbb{Q}[t]$. In sage I do the following:

K = PolynomialRing(QQ, ["t"])
R = PolynomialRing(K, ["x"])

t = K.gen(0)                                                                                             
x = R.gen(0)                                                                                             

f = (t**2 - QQ(1/10)*t + 1)*x**2 + (QQ(3/4)*t + QQ(7/2))*x - t + 8
f1 = f.specialization({t: 1})

This works fine and as expected $f_1$ is a polynomial only in $x$:

f1.parent() == QQ["x"] # True

Now I want to do exactly the same but over $\overline{\mathbb{Q}}$ instead:

L = PolynomialRing(QQbar, ["t"])
S = PolynomialRing(L, ["x"])

t = L.gen(0)
x = S.gen(0)

g = (t**2 - QQ(1/10)*t + 1)*x**2 + (QQ(3/4)*t + QQ(7/2))*x - t + 8
g1 = g.specialization({t: 1})

I would expect $g_1$ to be a polynomial only in $x$ as above, i.e. I would expect $g_1 \in \overline{\mathbb{Q}}[x]$. However, I get:

g1.parent() == QQbar["x"] # False
g1.parent() == S # True

Is this a bug? Or am I misunderstanding something?

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I do not know why SageMath does not reduce the parent ring upon specialization, but it can be enforced either by explicit conversion g1 = QQbar["x"](g1) or by defining a polynomial ring in two variables from the beginning:

LS = PolynomialRing(QQbar, ["t","x"])
t, x = LS.gens()
Max Alekseyev gravatar imageMax Alekseyev ( 2021-02-02 10:18:36 -0600 )edit