Program(s) for expressing a polynomial $f\in K[x_1,\cdots,x_n]$ in terms of generators of the ideal $\langle f_1,\cdots,f_r\rangle$

Question

Let $\mathbb{K}$ be field ($\mathbb{K}\in \{\mathbb{Q},\mathbb{R},\mathbb{C}\}$) and $K[x_1,\cdots,x_n]$ be the polynomials ring over $\mathbb{K}$.

Consider a list of polynomials $\{f_1,\cdots,f_r\}$ and a polynomial $f\in K[x_1,\cdots,x_n]$. Let $\langle f_1,\cdots,f_r\rangle$ be the ideal generated by $\{f_1,\cdots,f_r\}$.

I am currently using wxmaxima. There is a command to find out whether $f\in \langle f_1,\cdots,f_r\rangle$.

1. If $f\in \langle f_1,\cdots,f_r\rangle$, I would like to know whether there is any program (prefer open source program) to find out $g_1,\cdots,g_r\in K[x_1,\cdots,x_n]$ such that $f=g_1f_1+\cdots+g_rf_r$.
2. If we can find out $g_1,\cdots,g_r\in K[x_1,\cdots,x_n]$, can we request those $g_1,\cdots,g_r\in K[x_1,\cdots,x_n]$ are minimal in the sense that their overall degrees is as small as possible?

According to this post: (radical membership and ideal membership), we can also determine whether $f\in \sqrt{\langle f_1,\cdots,f_r\rangle}$ with the help of Gröbner basis.

If $f\in \sqrt{\langle f_1,\cdots,f_r\rangle}$, I would like to know whether there is any program (prefer open source program) to find out $m\in\mathbb{N}$ and $h_1,\cdots,h_r\in K[x_1,\cdots,x_n]$ such that $f^m=h_1f_1+\cdots+h_rf_r$.

If we can find out $m\in\mathbb{N}$ and $h_1,\cdots,h_r\in K[x_1,\cdots,x_n]$, can we request the following: (a) those $h_1,\cdots,h_r\in K[x_1,\cdots,x_n]$ are minimal in the sense that their overall degrees is as small as possible? (b) m is as small as possible?