1 | initial version |

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

The product `a1*x`

is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

The first approach is to let Sage consider `a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring to be with coefficients of the polynomial ring `R`

:

```
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and `5`

.

Another approach is to let the coefficients be elements of another polynomial ring, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

Now, the common zeroes of the coeficients of `Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

2 | No.2 Revision |

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

The product `a1*x`

is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

The first approach is to let Sage consider `a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring to be with coefficients of the polynomial ring `R`

:

```
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and

.~~5~~5-

Another approach is to let the coefficients be elements of another polynomial ring, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

Now, the common zeroes of the coeficients of `Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

3 | No.3 Revision |

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

The product `a1*x`

is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

The first approach is to let Sage consider `a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring to be with coefficients of the polynomial ring `R`

:

```
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and

.~~5-~~-5

Another approach is to let the coefficients be elements of another polynomial ring, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

Now, the common zeroes of the coeficients of `Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

4 | No.4 Revision |

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

`a1*x`

is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

`a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring to be with coefficients of the polynomial ring `R`

:

```
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and `-5`

.

Another approach is to let the coefficients be elements of another polynomial ~~ring, ~~ring `S`

, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

`Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

5 | No.5 Revision |

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

`a1*x`

is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

The first approach is to let Sage consider `a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring `R`

to be ~~with coefficients of the polynomial ring ~~`R`

`defined over the symbolic ring `

`SR`

:

` ````
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and `-5`

.

Another approach is to let the coefficients be elements of another polynomial ring `S`

, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

Now, the common zeroes of the coeficients of `Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

` `

` ` 6 No.6 Revision

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

The product `a1*x`

(and the other elementary operations) is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

The first approach is to let Sage consider `a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring `R`

to be defined over the symbolic ring `SR`

:

```
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and `-5`

.

Another approach is to let the coefficients be elements of another polynomial ring `S`

, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

Now, the common zeroes of the coeficients of `Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

7 No.7 Revision

Whe you write :

```
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
```

The product `a1*x`

(and the other elementary operations) is done using *coercion*, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in `x,y`

. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:

```
sage: Pol.parent()
Symbolic Ring
```

In particular, you lose the polynomial structure in `x,y`

.

The first approach is to let Sage consider `a0`

and `a1`

a possible coefficients for polynomials in `x,y`

, by declaring the polynomial ring `R`

to be defined over the symbolic ring `SR`

:

```
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
```

Then you can extract the nonzero coefficients:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
```

There is of course no solution since `a0`

should be both equal to `0`

and `-5`

.

Another approach is to let the coefficients be elements of another polynomial ring `S`

, that will be the ring over which `R`

will be defined:

```
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
```

Now, the common zeroes of the coeficients of `Pol`

(which are polynomials, elements of `S`

), is nothing but the variety of an ideal:

```
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
```

Which is empty as well.

` `

` `

```
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.
```