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The resultant $\operatorname{Res}(f,g)$ of two polynomials over a field $k$ is a polynomial in the coefficients of $f$ and $g$ which enjoys the property of being nonzero if and only if $f$ and $g$ have no common root in an algebraic closure $\overline{k}$ of $k$.

Does there exist a similar construction for three polynomials? There seems to be none. I would like to suggest the following conjecture:

Conjecture: there does not exist a function $\operatorname{Res}(f,g,h)$ of three polynomials $f,g,h \in k[x]$, which is a polynomial in the coefficients of $f,g,h$, having the property of being zero if and only if the polynomials $f,g,h$ have a common root in an algebraic closure $\overline{k}$ of $k$.

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up vote 10 down vote accepted

This can be done using resolvants - which date back to Kronecker if not earlier. Below is an elementary introduction from p.14 of Elkadi et al. Algebraic geometry and geometric modeling. enter image description here enter image description here

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That's great! Kronecker answers my question from the grave. – Bruno Joyal Jun 24 '11 at 1:41

There is a resultant for three polynomials in two variables. It's considerably trickier than the resultant for two polynomials in one variable.

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Would you mind elaborating a bit? I'm really talking about one variable though. :) – Bruno Joyal Jun 24 '11 at 0:48
@Bruno Google "Dixon resultant". – Bill Dubuque Jun 24 '11 at 1:02

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