I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a homogeneous variety. My first idea is to compute the radical of the ideal $I$ that they generate, and then check whether the homogeneous components of each generator of $\sqrt{I}$ are in $\sqrt{I}$. Is there a more efficient way to do this?
For context: I have 96 degree-3 polynomials in 16 variables. I'm not hopeful that Macaulay2 will compute $\sqrt{I}$ in any reasonable amount of time (it has been running for 24 hours already).
Edit: Thank you, KReiser for suggesting I check if $I$ itself is homogeneous by checking if homogeneous components of the generators are in $I$.
Edit 2: I have now verified in Macaulay2 that $I$ itself is not homogeneous, so I am left with the original question of how to efficiently compute whether $\sqrt{I}$ is homogeneous.
Here's some more information on my specific situation. Let's arrange the 16 variables into a $4 \times 4$ matrix
$$X= \begin{bmatrix} A && B \\ C && D\end{bmatrix}. $$
My polynomials are the $3 \times 3$ minors of $X$, the $3 \times 3$ minors of $$ \begin{bmatrix} A^T && B^T \\ C^T && D^T\end{bmatrix}. $$
and the $3 \times 3$ minors of
$$ \begin{bmatrix} \vec{A} && \vec{B} && \vec{C} && \vec{D}\end{bmatrix}, $$
where $\vec{A}$, etc. is the $2 \times 2$ matrix $A$ viewed as a $4 \times 1$ column vector. This gives $3*{4 \choose 3}^2=48$ polynomials (which generate a homogeneous radical ideal). The other 48 polynomials are the same, with $X$ replaced by
$$ \begin{bmatrix} 0&&1&&1&&0\\1&&0&&0&&0\\1&&0&&0&&0\\0&&0&&0&&0\end{bmatrix}+ \begin{bmatrix} A && B \\ C && D\end{bmatrix}. $$
The ideal generated by these latter 48 polynomials (call it $J$) is certainly not homogeneous, but I'm not sure about $I$.