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Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is an ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is an ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators.

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is an ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators.

Edit3: In case it is helpful, the case I'm looking at is a ring with an action of a group, and its subring of invariants for which I have generators.

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is an ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators.

Edit3: In case it is helpful, the case I'm looking at is a ring with an action of a group, and its subring of invariants for which I have generators.

Edit4: Here's a suggested algorithm: sort the generators by degree. Then sequentially add them, and check if they were already in the subring. Does Sage even have suitable subring capabilities?

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is an ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators.generators and it is those 2 generators that I want.

Edit3: In case it is helpful, the case I'm looking at is a ring with an action of a group, and its subring of invariants for which I have generators.

Edit4: Here's a suggested algorithm: sort the generators by degree. Then sequentially add them, and check if they were already in the subring. Does Sage even have suitable subring capabilities?

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is an a homogeneous ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators and it is those 2 generators that I want.

Edit3: In case it is helpful, the case I'm looking at is a ring with an action of a group, and its subring of invariants for which I have generators.

Edit4: Here's a suggested algorithm: sort the generators by degree. Then sequentially add them, and check if they were already in the subring. Does Sage even have suitable subring capabilities?

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ (say $\mathbb{Q}$) and a list of polynomials $p_1,..,p_n$ in $R$, is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is a homogeneous ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators and it is those 2 generators that I want.

Edit3: In case it is helpful, the case I'm looking at is a ring with an action of a group, and its subring of invariants for which I have generators.

Edit4: Here's a suggested algorithm: sort the generators by degree. Then sequentially add them, and check if they were already in the subring. Does Sage even have suitable subring capabilities?

Compute minimal number of generators of subring

Hi all,

Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ (say $\mathbb{Q}$) and a list of polynomials $p_1,..,p_n$ in $R$, $R$ (homogeneous say), is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?

Next, suppose $I$ is a homogeneous ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?

Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators and it is those 2 generators that I want.