1 | initial version |

You can use `PolynomialRing`

in the following way.

```
sage: R.<d,e> = PolynomialRing(QQ)
sage: S.<b,c> = PolynomialRing(R)
sage: S
Multivariate Polynomial Ring in b, c over Multivariate Polynomial Ring in d, e over Rational Field
sage: var('b c d e')
(b, c, d, e)
sage: a = b*d^2+c*e^2+e*(b+2*c*d)
sage: a
b*d^2 + c*e^2 + (2*c*d + b)*e
sage: S(a)
(d^2 + e)*b + (2*d*e + e^2)*c
```

2 | No.2 Revision |

To work with variables b_i and x_i as you want, define multivariate polynomial rings Rb and Rx as in the code below, then inject variables as shown. You can ~~use ~~then enter any polynomial expressions in the `PolynomialRing`

~~following way.~~b_i's and x_i's, and it will be understood as an element in Rb if no x_i's are involved, and otherwise in Rx, ie, as a polynomial in the x_i's with coefficients in polynomials in the b_i's.

`sage: `~~R.<d,e> ~~Rb = ~~PolynomialRing(QQ)
~~PolynomialRing(ZZ,10,name='b')
sage: ~~S.<b,c> ~~Rx = ~~PolynomialRing(R)
~~PolynomialRing(Rb,10,name='x')
sage: ~~S
Multivariate Polynomial Ring in b, c over Multivariate Polynomial Ring in d, e over Rational Field
~~Rb.inject_variables()
Defining b0, b1, b2, b3, b4, b5, b6, b7, b8, b9
sage: ~~var('b c d e')
(b, c, d, e)
~~Rx.inject_variables()
Defining x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
sage: ~~a = b*d^2+c*e^2+e*(b+2*c*d)
sage: a
b*d^2 ~~b1 * (x2 + ~~c*e^2 ~~x3) + ~~(2*c*d ~~b2 * (x3 + ~~b)*e
sage: S(a)
(d^2 ~~x8)
b1*x2 + ~~e)*b ~~(b1 + ~~(2*d*e ~~b2)*x3 + ~~e^2)*c
~~b2*x8

3 | No.3 Revision |

~~To ~~The best way to work with variables b_i and x_i as you ~~want, ~~want
is to define multivariate polynomial rings Rb and Rx ~~as in the code below, then inject variables as shown. You can then enter ~~and inject
variables. Then any polynomial ~~expressions ~~expression
in the b_i's and ~~x_i's, and it ~~x_i's will be ~~understood ~~transformed as ~~an element in Rb if no x_i's are involved, and otherwise in Rx, ie, as a polynomial ~~a polynomial
in the x_i's with coefficients in polynomials in the b_i's.

```
sage: Rb = PolynomialRing(ZZ,10,name='b')
sage: Rx = PolynomialRing(Rb,10,name='x')
sage: Rb.inject_variables()
Defining b0, b1, b2, b3, b4, b5, b6, b7, b8, b9
sage: Rx.inject_variables()
Defining x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
sage: b1 * (x2 + x3) + b2 * (x3 + x8)
b1*x2 + (b1 + b2)*x3 + b2*x8
```

For your baby example, you could work in the symbolic ring and just ask coefficients,

```
sage: var('b c d e')
(b, c, d, e)
sage: a = b*d^2+c*e^2+e*(b+2*c*d)
sage: a
b*d^2 + c*e^2 + (2*c*d + b)*e
sage: a.coefficient(b)
d^2 + e
sage: a.coefficient(c)
2*d*e + e^2
```

or you could mix the two in the following way.

```
sage: S.<d,e> = PolynomialRing(QQ)
sage: R.<b,c> = PolynomialRing(S)
sage: R
Multivariate Polynomial Ring in b, c over Multivariate Polynomial Ring in d, e over Rational Field
sage: var('b c d e')
(b, c, d, e)
sage: a = b*d^2+c*e^2+e*(b+2*c*d)
sage: a
b*d^2 + c*e^2 + (2*c*d + b)*e
sage: R(a)
(d^2 + e)*b + (2*d*e + e^2)*c
```

But as a rule, it is best to avoid the symbolic ring altogether, whenever possible. And the way your main question is formulated, the most sensible approach is the polynomial rings one.

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