Suppose $f$ is a degree $n$ univariate polynomial with roots $\alpha_1, \ldots, \alpha_n$. Then we know that \frac{f'(x)}{f(x)} = \sum_{i=1}^n \frac{1}{(x-\alpha_i)}. Can we say something similar for a system $F=(f,g)$, where $f,g$ are bivariate polynomials with only finitely many common zeros? That is, can we express F'/F in a form dependent upon the common roots of $f,g$?
Zeros of a system of bivariate polynomials
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0You could probably work out something per coordinate (calculate $F \cap K[x]$ and $F \cap K[y]$), but I am not aware of any formula that gives direct results. Why exactly do you need such a correspondence? – 2012-03-14