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In previous work of the authors and their collaborators (see Progress inMath, vol. 114, Birk\"auser, 1993) it was shown how the equivalence of severalconstructions of residue currents associated to complete intersection familiesof (germs of) holomorphic functions in ${\bf C}^n$ could be profitably used tosolve algebraic problems like effective versions of the Nullstellensatz. Inthis work we explain how an application of similar ideas in the non-completeintersection case leads to a remarkable algebraic result, namely:  Let $P_1,...,P_n$ be $n$ polynomials in $n$ variables such that the zero setof $P_1,...,P_n$ 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 idealgenerated by the $P_j$, $j=1,...,n$.  The same methods lead to further insights into the construction of Greencurrents associated to effective cycles in projective space.

Analytic residue theory in the non-complete intersection case