1 | initial version |

First, it makes sense that you will get `[1.0]`

as the only element of the Groebner basis since, as the polynomial `sum_sqr_distances`

has coefficients in the inexact ring `RR`

, it is likely that the given polynomials generate the whole polynomial ring.

Indeed, when you add a polynomial with coefifients in `RR`

with polynomials with coefficients in `QQ`

, the result will belong to a polynomial ring with coefficients in `RR`

. You can check it by typing:

sage: I ..... over Real Field with 53 bits of precision

You can replace `QQ`

with `RDF`

in the definition of the polynomial ring. Then the coercion will be done towards `RDF`

which consist of low-level floating-point numbers, hence the computation will finish, while it might take longer with `RR`

.

However, i guess that having `[1.0]`

as the only element of the Groebner basis due to working in an inexact ring is quite unsatisfactory.

So, if you want to work on an exact ring, since the name `sum_sqr_distances`

seems to indicate that the coefficients are quadratic numbers, you should define them either in `QQbar`

or in some number field. It is likely that the computation of a Groebner basis will be even slower.

2 | No.2 Revision |

First, it makes sense that you will get `[1.0]`

`the coinstant polynomial `

`1.0`

as the only element of the Groebner basis since, as the polynomial `sum_sqr_distances`

has coefficients in the inexact ring `RR`

, it is likely that the given polynomials generate the whole polynomial ring.

` `Indeed, when you add a polynomial with coefifients in `RR`

with polynomials with coefficients in `QQ`

, the result will belong to a polynomial ring with coefficients in `RR`

. You can check it by typing:

```
sage: I
..... over Real Field with 53 bits of
```~~precision~~precision

You can replace `QQ`

with `RDF`

in the definition of the polynomial ring. Then the coercion will be done towards `RDF`

which consist of low-level floating-point numbers, hence the computation will finish, while it might take longer with `RR`

.

However, i guess that having `[1.0]`

as the only element of the Groebner basis due to working in an inexact ring is quite unsatisfactory.

So, if you want to work on an exact ring, since the name `sum_sqr_distances`

seems to indicate that the coefficients are quadratic numbers, you should define them either in `QQbar`

or in some number field. It is likely that the computation of a Groebner basis will be even slower.

` `

` ` 3 No.3 Revision

First, it makes sense that you will get the coinstant polynomial `1.0`

as the only element of the Groebner basis since, as the polynomial `sum_sqr_distances`

has coefficients in the inexact ring `RR`

, it is likely that the given polynomials generate the whole polynomial ring.

Indeed, when you add a polynomial with coefifients in `RR`

with polynomials with coefficients in `QQ`

, the result will belong to a polynomial ring with coefficients in `RR`

. You can check it by typing:

```
sage: I
..... over Real Field with 53 bits of precision
```

You can replace `QQ`

with `RDF`

in the definition of the polynomial ring. Then the coercion will be done towards `RDF`

which consist of low-level floating-point numbers, hence the computation will finish, while it might take longer with `RR`

.

However, i guess that having `[1.0]`

as the only element of the Groebner basis due to working in an inexact ring is quite unsatisfactory.

So, if you want to work on an exact ring, since the name `sum_sqr_distances`

seems to indicate that the coefficients are quadratic numbers, you should define them either in `QQbar`

or in some number field. It is likely that the computation of a Groebner basis will be even slower.

Of course, you can convert `sum_sqr_distances`

into a polynomial with rational coefficients as follows:

```
sage: sum_sqr_distances = P(sum_sqr_distances)
```

But regarding the "sqr" name of the polynomial, it is likely not to be the way to go.

4 No.4 Revision

First, it makes sense that you will get the coinstant polynomial `1.0`

as the only element of the Groebner basis since, as the polynomial `sum_sqr_distances`

has coefficients in the inexact ring `RR`

, it is likely that the given polynomials generate the whole polynomial ring.

Indeed, when you add a polynomial with coefifients in `RR`

with polynomials with coefficients in `QQ`

, the result will belong to a polynomial ring with coefficients in `RR`

. You can check it by typing:

```
sage: I
..... over Real Field with 53 bits of precision
```

You can replace `QQ`

with `RDF`

in the definition of the polynomial ring. Then the coercion will be done towards `RDF`

which consist of low-level floating-point numbers, hence the computation will finish, while it might take longer with `RR`

.

However, i guess that having `[1.0]`

as the only element of the Groebner basis due to working in an inexact ring is quite unsatisfactory.

So, if you want to work on an exact ring, since the name `sum_sqr_distances`

seems to indicate that the coefficients are quadratic numbers, you should define them either in `QQbar`

or in some number field. It is likely that the computation of a Groebner basis will be even slower.

Of course, you can convert `sum_sqr_distances`

into a polynomial with rational coefficients as follows:

```
sage: sum_sqr_distances = P(sum_sqr_distances)
```

But regarding the "sqr" name of the polynomial, it is likely not to be the way to go.

Now, regarding your initial problem, perhaps should you forget about polynomial equations, instead, look at the optimization features of Sage, in particular, you can have a look at:

```
sage: minimize?
```

5 No.5 Revision

First, it makes sense that you will get the coinstant polynomial `1.0`

as the only element of the Groebner basis since, as the polynomial `sum_sqr_distances`

has coefficients in the inexact ring `RR`

, it is likely that the given polynomials generate the whole polynomial ring.

Indeed, when you add a polynomial with coefifients in `RR`

with polynomials with coefficients in `QQ`

, the result will belong to a polynomial ring with coefficients in `RR`

. You can check it by typing:

```
sage: I
..... over Real Field with 53 bits of precision
```

You can replace `QQ`

with `RDF`

in the definition of the polynomial ring. Then the coercion will be done towards `RDF`

which consist of low-level floating-point numbers, hence the computation will finish, while it might take longer with `RR`

.

However, i guess that having `[1.0]`

as the only element of the Groebner basis due to working in an inexact ring is quite unsatisfactory.

So, if you want to work on an exact ring, since the name `sum_sqr_distances`

seems to indicate that the coefficients are quadratic numbers, you should define them either in `QQbar`

or in some number field. It is likely that the computation of a Groebner basis will be even slower.

Of course, you can convert `sum_sqr_distances`

into a polynomial with rational coefficients as follows:

```
sage: sum_sqr_distances = P(sum_sqr_distances)
```

But regarding the "sqr" name of the polynomial, it is likely not to be the way to go.

Now, regarding your initial problem, perhaps should you forget about polynomial equations, instead, look at the optimization features of Sage, in particular, you can have a look at:

```
sage: minimize?
```

You can also have a look at this question

` `

` `

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