Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $k$.

share|improve this question

2 Answers 2

up vote 9 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

share|improve this answer
1  
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.

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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.