Applying group theory to multivariate eqs

[Stephen Tashi]

Are there any good examples of how group theory can be applied to solve multivariate algebraic equations?

The type of equations I have in mind are those that set a "multilinear" polynomial (e.g. $xyz + 3xy + z$) equal to a monomial (e.g. $x^3$). However, I'd like to hear about any sort of simultaneous algebraic equations where group theory is useful.

 

[micromass]

Group theory? I don't think it's very useful. You should read up about elimination theory and Grobner bases though.

 

[Stephen Tashi]

Group theory? I don't think it's very useful.

Are we barking up the wrong tree in post #33 of https://www.physicsforums.com/threa...ible-with-radicals.895637/page-2#post-5654053 ?

 

[fresh_42]

Group theory? I don't think it's very useful. You should read up about elimination theory and Grobner bases though.

Well, at least there are some applications for half-groups in social sciences (e.g. degree of relationships), finite groups in cryptography (e.g. error correcting codes), chemistry (geometry of molecules) and crystallography (symmetry groups) and for infinite groups in physics (e.g. QFT, Emmy Noether's theorem (she explicitly mentiones Lie's work in her papers)).

 

[micromass]

Well, at least there are some applications for half-groups in social sciences (e.g. degree of relationships), finite groups in cryptography (e.g. error correcting codes), chemistry (geometry of molecules) and crystallography (symmetry groups) and for infinite groups in physics (e.g. QFT, Emmy Noether's theorem (she explicitly mentiones Lie's work in her papers)).

Of course, I never said group theory was useless. I just said that for this particular question, it's not really useful.

Of course, the question in the OP is vague. But you might for example extend it to "find all rational solutions of $y^2 = x^2 + x + 1$", then group theory becomes a lot more useful.