1 | initial version |

One possibility is to use multivariate Laurent polynomials and let one variable be your parameter

```
sage: R.<a,t> = LaurentPolynomialRing(QQ)
sage: R
Multivariate Laurent Polynomial Ring in a, t over Rational Field
```

another one is to use coefficient being themselves polynomials

```
sage: P.<t> = PolynomialRing(QQ)
sage: R.<a> = LaurentPolynomialRing(P)
sage: R
Univariate Laurent Polynomial Ring in a over Univariate Polynomial Ring in t over Rational Field
```

There are also several other ways but everything depend on what you want to do with them... try to make your question more precise.

2 | No.2 Revision |

One possibility is to use multivariate Laurent polynomials and let one variable be your parameter

`sage: R.<a,t> = `~~LaurentPolynomialRing(QQ)
~~LaurentSeriesRing(QQ)
sage: R
Multivariate Laurent Polynomial Ring in a, t over Rational Field

another one is to use coefficient being themselves polynomials

```
sage: P.<t> = PolynomialRing(QQ)
sage: R.<a> =
```~~LaurentPolynomialRing(P)
~~LaurentSeriesRing(P)
sage: R
Univariate Laurent Polynomial Ring in a over Univariate Polynomial Ring in t over Rational Field

There are also several other ways but everything depend on what you want to do with them... try to make your question more precise.

EDIT: to expend a quotient (of polynomials) with parameters, the second method is more appropriate

```
sage: P.<t> = PolynomialRing(QQ)
sage: R.<a> = LaurentSeriesRing(P)
sage: num = (t^2 + 1) * a^2 + (-t + 2) * a + (t^2 - 1)
sage: den = (t^3 + t + 1) * a^2 + (2*t + 5) * a - 1
sage: num / den
(-t^2 + 1) + (-2*t^3 - 5*t^2 + 3*t + 3)*a + ... + O(a^20)
```

3 | No.3 Revision |

One possibility is to use multivariate Laurent polynomials and let one variable be your parameter

```
sage: R.<a,t> = LaurentSeriesRing(QQ)
sage: R
Multivariate Laurent Polynomial Ring in a, t over Rational Field
```

another one is to use ~~coefficient ~~coefficients being themselves polynomials

```
sage: P.<t> = PolynomialRing(QQ)
sage: R.<a> = LaurentSeriesRing(P)
sage: R
Univariate Laurent Polynomial Ring in a over Univariate Polynomial Ring in t over Rational Field
```

There are also several other ways but everything ~~depend ~~depends on what you want to do with them... try to make your question more precise.

EDIT: to ~~expend ~~expand a quotient (of polynomials) with parameters, the second method is more appropriate

```
sage: P.<t> = PolynomialRing(QQ)
sage: R.<a> = LaurentSeriesRing(P)
sage: num = (t^2 + 1) * a^2 + (-t + 2) * a + (t^2 - 1)
sage: den = (t^3 + t + 1) * a^2 + (2*t + 5) * a - 1
sage: num / den
(-t^2 + 1) + (-2*t^3 - 5*t^2 + 3*t + 3)*a + ... + O(a^20)
```

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