How to find a finite set of generators for $I \subset k[x_1, ..., x_n]$

Suppose that we have a set of polynomials $f_1, ..., f_m \in C = k[x_1, ..., x_n]$ where $k$ is an algebraically closed field, and let $V$ denote the set on which they all simultaneously vanish. Define $I(V)$ to be the ideal of polynomials in $C$ which vanish on all of $V$.

In general, $I(V) = \text{rad}(f_1, ..., f_m)$ by Hilbert's nullstellensatz, and it is finitely generated by the Hilbert's basis theorem, but I was wondering is there a good way to find a set of generators for $I(V)$? Essentially I am asking, is there a good way to find the generators of the radical of an ideal, given that ideal's generators?

I am interested in this problem, because I am interested in solving systems of polynomial equations in several variables. I was thinking I could use Grobner bases to reduce a system of equations to a simpler system involving less variables, but as far as I'm aware in order to computer a Grobner basis one needs a generating set already. It could be that I am lucky and $I(V) = (f_1, ..., f_m)$, but that seems quite rare to me!

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2022-07-25 20:47:06
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