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Let $k$ be a field, and $f:k[x_1,\ldots,x_n]\to k[y_1,\ldots,y_m]$ a $k$-algebra homomorphism. Given $r_1,\ldots,r_k\in k[y_1,\ldots,y_m]$, is there an algorithm for producing a finite generating set for the ideal $f^{-1}((r_1,\ldots,r_k))$?

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I wonder if there is even a reasonable algorithm for finding generators of $f^{-1}(0)$. – Thomas Andrews Jul 18 '13 at 16:42
Will Grobner bases help? – Robert Lewis Jul 18 '13 at 17:37

The answer is yes. The question is asking to compute the kernel of

$$f : k[x_1, \dots, x_n] \rightarrow S$$

where $S$ is some quotient of a polynomial ring. Macaulay2 can calculate such a kernel.

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