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Let $I$ be a graded ideal in a polynomial ring $R$, which is generated minimally by $x_1,...,x_k$. Then the power of $I$, i.e $I^t$ is generated by monomials of the form $x_{1}^{a_1}...x_{n}^{a_{n}}$ where $a_1+...+a_n=t$. Denote this set by $S$.

Can we say anything (others than above)about the minimal generating set of $I^{t}$? Is it $S$ ?

Given a minimal generating set for a graded ideal in a graded commutative ring, from these how much do we know about the minimal generating set for the power of it?

Edit : Here is an example for precising my question :

In the polynomial ring $k[x,y,z]$ let $I=(x^2, xy^3, y^2z^3)$, then $I^2=(x^4, x^2y^6, y^4z^6, x^3y^3, x^3y^5z^3xy^5z^3, x^2y^2z^3)$

Is $\lbrace x^4, x^2y^6, y^4z^6, x^3y^3, x^3y^5z^3xy^5z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?

Update There are some typing mistake that I have not noticed. I have change my question. This time, the generating set of $I$ is minimal. So what can we say about the generating set for $I^2$ above ? Is it minimal? Thank you everyone for helping me answer my question!

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Let $I$ be a graded ideal in a polynomial ring $R$, which is generated minimally by $x_1,...,x_k$. Then the power of $I$, i.e $I^t$ is generated by monomials of the form $x_{1}^{a_1}...x_{n}^{a_{n}}$ where $a_1+...+a_n=t$. Denote this set by $S$.

Can we say anything (others than above)about the minimal generating set of $I^{t}$? Is it $S$ ?

Given a minimal generating set for a graded ideal in a graded commutative ring, from these how much do we know about the minimal generating set for the power of it?

Edit : Here is an example for precising my question :

In the polynomial ring $k[x,y,z]$ let $I=(x^2, x^3yxy^3, y^2z^3)$, then $I^2=(x^4, x^6y^2x^2y^6, y^4z^6, x^5yx^3y^3, x^3y^3z^3x^3y^5z^3, x^2y^2z^3)$

Is $\lbrace x^4, x^6y^2x^2y^6, y^4z^6, x^5yx^3y^3, x^3y^3z^3x^3y^5z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?

Update As Mariano Suarez Alvarez pointed out, we do There are some typing mistake that I have not need $x^6y^2$ since we noticed. I have $x^4$. But what about $x^5y$, since $x^4\times xy=x^5y$change my question. This time, and for $x^3y^3z^3$, the generating set of $x^2y^2z^3$. I$is minimal. So , what can I talk we say about the minimal generating set for$I^2$now above ? Is it minimal? Thank you everyone for helping me answer my question! 5 deleted 8 characters in body Let$I$be a graded ideal in a graded commutative polynomial ring$R$, which is generated minimally by$x_1,...,x_k$. Then the power of$I$, i.e$I^t$is generated by monomials of the form$x_{1}^{a_1}...x_{n}^{a_{n}}$where$a_1+...+a_n=t$. Denote this set by$S$. Can we say anything (others than above)about the minimal generating set of$I^{t}$? Is it$S$? Given a minimal generating set for a graded ideal in a graded commutative ring, from these how much do we know about the minimal generating set for the power of it? Edit : Here is an example for precising my question : In the polynomial ring$k[x,y,z]$let$I=(x^2, x^3y, y^2z^3)$, then$I^2=(x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3)$Is$\lbrace x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3\rbrace$a minimal generating set for$I^2$? Update As Mariano Suarez Alvarez pointed out, we do not need$x^6y^2$since we have$x^4$. But what about$x^5y$, since$x^4\times xy=x^5y$, and for$x^3y^3z^3$,$x^2y^2z^3$. So, what can I talk about the minimal generating set for$I^2\$ now ? Thank you everyone for helping me answer my question!

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