Converting polynomials between rings

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In your example, having defined

sage: R1.<a,b,c,t> = PolynomialRing(QQ)
sage: L = (a*t^2+b*t+c).subs(a=2,b=3,c=4)

we get

sage: L
2*t^2 + 3*t + 4

(did you mean L = a*t^2+b*t+c without substituting values for a, b, c?)

Having then defined

sage: R2.<A,B,C,D,t> = PolynomialRing(QQ)
sage: p = (A*t+B)^2+(C*t+D)^2

you can convert polynomials from one ring to the other

sage: R2(L)
2*D^2 + 3*D + 4

Here, the order of the variables matters, more than their names: t, the fourth variable in R1, is mapped to D, the fourth variable in R2.

If you want a, b,c, t to be mapped to A, B, C, t, you might want to include an extra variable d in R1 and not use it.

Or you could compare string representations of your polynomials, applying string replacements as necessary. Or you could use p.monomials() and p.coefficients().

You could also define a ring homomorphism from R1 to R2 mapping the variables to the variables of your choice, see the reference manual.

Because of the order you've defined the objects, L(t=t) will be an element of R2. Slightly more robustly, in your example you can do any of the following.

sage: L(t=R2.gen(4)).parent()
Multivariate Polynomial Ring in A, B, C, D, t over Rational Field
sage: L(t=R2.4).parent()
Multivariate Polynomial Ring in A, B, C, D, t over Rational Field