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Let $G = \{g_1, g_2, \ldots, g_n\}$ be a finite set of polynomials in $\mathbb{F}[x_1, x_2, \ldots, x_m]$. I'm interested in polynomial dependencies among these polynomials. More specifically, polynomials $f \in \mathbb{F}[G]$ such that $f(g_1, g_2, \ldots, g_n) = 0$. Are there any well known results relating to this kind of thing? Really I'm just looking for directions on things to google. I suspect this might be related to Algebraic Geometry, but I don't know Algebraic Geometry so I'm kind of lost. Thanks for any help!

Nothingisreallyworking
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1 Answers1

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The topic you describe belongs to the area called elimination theory. If you want to find all the relations as described in your post, you're looking for the intersection of the ideal $(y_i-g_i)\subset \Bbb F[x_1,\cdots,x_m,y_1,\cdots,y_n]$ with the subring $\Bbb F[y_1,\cdots,y_n]$. This is called computing an elimination ideal, and the main approach to compute this uses Grobner bases, which is a more sophisticated way of doing polynomial long division in the multivariate setting by playing around with the idea of "leading terms".

This can be somewhat computationally intensive, so frequently it's in your best interest to get a computer to do it for you. Many computer algebra systems have an implementation of this algorithm - for instance, Sage will, with online cell here and documentation about elimination ideals here.

KReiser
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