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# Calculatingbi-homogenizationAnyimplementedalgorithmtocomputetheclosure of an idealaffinevarietyinaproductofprojectivespaces?

Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-homogenization' $\tilde I$ of $I$ with respect to $x$ and $y$ variables?

(Recall that the 'bi-homogenization' of a polynomial $f = \sum a_{\alpha, \beta} x^\alpha y^\beta$ is by definition $\tilde f := \sum a_{\alpha, \beta} x^\alpha y^\beta x_0^{d - |\alpha|} y_0^{e- |\beta|}$, where $x_0$ and $y_0$ are two new variables, $d := \deg_x(f)$ and $e := \deg_y(f)$. Then $\tilde I :=${$\tilde f: f \in I$}.)

I can think

My motivation is geometric: to find the closure $\overset{-}{V}$ of a way subvariety $V$ of $k^{m+n}$ in $\mathbb{P}^m \times \mathbb{P}^n$. Of course I could as well calculate the Segre embedding of $\overset{-}{V}$ in $\mathbb{P}^{mn + m +n}$, but I would like to do ithave something computationally less expensive.It

I can think of an algorithm which involves introducing $n$ (or $m$, whichever is the smaller) new variables $t_1, \ldots, t_n$ and computing the monomial basis of an ideal $J$ in $k[x,y,t]$, where $J$ is to be constructed from $I$. But I was wondering if someone had already implemented some (possibly better) algorithm which would do this job.

My motivation is geometric: to find the closure $\overset{-}{V}$ of a subvariety $V$ of $k^{m+n}$ in $\mathbb{P}^m \times \mathbb{P}^n$. Of course I could as well calculate the Segre embedding of $\overset{-}{V}$ in $\mathbb{P}^{mn + m +n}$, but I would like to have something computationally less expensive.

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# Calculating bi-homogenization of an ideal

Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-homogenization' $\tilde I$ of $I$ with respect to $x$ and $y$ variables? (Recall that the 'bi-homogenization' of a polynomial $f = \sum a_{\alpha, \beta} x^\alpha y^\beta$ is by definition $\tilde f := \sum a_{\alpha, \beta} x^\alpha y^\beta x_0^{d - |\alpha|} y_0^{e- |\beta|}$, where $x_0$ and $y_0$ are two new variables, $d := \deg_x(f)$ and $e := \deg_y(f)$. Then $\tilde I :=${$\tilde f: f \in I$}.)

I can think of a way to do it. It involves introducing $n$ (or $m$, whichever is the smaller) new variables $t_1, \ldots, t_n$ and computing the monomial basis of an ideal $J$ in $k[x,y,t]$, where $J$ is to be constructed from $I$. But I was wondering if someone had already implemented some (possibly better) algorithm which would do this job.

My motivation is geometric: to find the closure $\overset{-}{V}$ of a subvariety $V$ of $k^{m+n}$ in $\mathbb{P}^m \times \mathbb{P}^n$. Of course I could as well calculate the Segre embedding of $\overset{-}{V}$ in $\mathbb{P}^{mn + m +n}$, but I would like to have something computationally less expensive.