Suppose that I have $n$ homogeneous polynomials $f\_1, \dots, f\_n \in \mathbb{C}[x\_1, \dots, x\_m]$ and that $n < m$. Is there a well known method or algorithm to determine if these polynomials are algebraically independent?


As far as I know the Jacobian criterion works only for the case where $n=m$.



How to show a set of polynomials is algebraically independent?
The polynomials are algebraically independent if and only if
$$
df\_1 \wedge df\_2 \wedge \cdots \wedge df\_n 
$$
is not identically zero. In other words, you have only to check that 
one of the maximal minors of the matrix $\left( \frac{\partial f\_i}{\partial x\_j} \right)$ is nonzero. 


