  In previous work of the authors and their collaborators (see Progress in
Math, vol. 114, Birk\"auser, 1993) it was shown how the equivalence of several
constructions of residue currents associated to complete intersection families
of (germs of) holomorphic functions in ${\bf C}^n$ could be profitably used to
solve algebraic problems like effective versions of the Nullstellensatz. In
this work we explain how an application of similar ideas in the non-complete
intersection case leads to a remarkable algebraic result, namely:
  Let $P_1,...,P_n$ be $n$ polynomials in $n$ variables such that the zero set
of $P_1,...,P_n$ can be defined as the zero set of $P_1,...,P_\nu$, with $\nu <
n$. Then, the Jacobian $J(P_1,...,P_n)$ of $(P_1,...,P_n)$ is in the ideal
generated by the $P_j$, $j=1,...,n$.
  The same methods lead to further insights into the construction of Green
currents associated to effective cycles in projective space.
