# “Resultant” of three polynomials

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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## 2 Answers

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.

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