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Given a set of generators for a polynomial ideal $I = <f_1, ..., f_n>$, is it possible to compute a set of generators for the radical $\sqrt I$ without first computing a Grobner basis for $I$?

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  • $\begingroup$ Do you have a specific ideal in mind? There are plenty of ideals $I$ where the radical can be computed without necessarily computing a Groebner basis, but are you looking for examples, or are you looking for a "general" result? $\endgroup$ Mar 23, 2014 at 2:22
  • $\begingroup$ I'm looking for a general result. $\endgroup$
    – Alex Flint
    Mar 23, 2014 at 2:55
  • $\begingroup$ How general? If the $f_i$ are monomials, there are simple, effective algorithms to directly compute the radical. If the $f_i$ form a regular sequence, at least it is simple to compute if the ideal is already radical. If you want something so general that it will apply to all ideals, then probably you need to use Groebner bases (or something very similar) to have an actual algorithm. $\endgroup$ Mar 23, 2014 at 15:54
  • $\begingroup$ @JasonStarr: Thanks, I think that answers my question (in the negative). I'm looking for something completely general. It would be nice if I could show that computing generators for the radical for $I$ is equivalent in some sense to computing the Grobner basis for $I$. $\endgroup$
    – Alex Flint
    Mar 23, 2014 at 23:27

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