# How to show a set of polynomials is algebraically independent?

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 $n=m$.

• You can consider the equations $f_i=t_i$ with new variables $t_i$, and eliminate the $x$'s using Groebner bases. Oct 8, 2010 at 20:57
• (This is explained, if I recall correctly, in Cox's book on Varieties, Ideals and algorithms) Oct 8, 2010 at 20:58

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.
• Martin, in char. 0 the map $f:\mathbf{A}^m \rightarrow \mathbf{A}^n$ is smooth on a dense open in the source if and only if it is dominant, and such generic smoothness is equivalent to generic injectivity of the induced map $f^{\ast}(\Omega^1_{\mathbf{A}^n/k}) \rightarrow \Omega^1_{\mathbf{A}^m/k}$. But at the generic point, such injectivity can be checked between $n$th exterior powers. Oct 9, 2010 at 12:47