Applying group theory to multivariate eqs

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Discussion Overview

The discussion revolves around the application of group theory to solve multivariate algebraic equations, particularly focusing on multilinear polynomials and their relation to monomials. Participants explore various mathematical frameworks and their relevance to the topic.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about examples of applying group theory to solve multivariate algebraic equations, specifically mentioning multilinear polynomials.
  • Another participant expresses skepticism about the usefulness of group theory for this purpose, suggesting that elimination theory and Grobner bases may be more relevant.
  • A repeated sentiment from multiple participants indicates doubt regarding the applicability of group theory, questioning whether it is the right approach for the problem at hand.
  • Some participants acknowledge that while group theory may not be useful for the original question, it has applications in other fields such as social sciences, cryptography, chemistry, and physics.
  • One participant suggests that the vagueness of the original question limits the applicability of group theory, proposing a more specific example where group theory could be beneficial.

Areas of Agreement / Disagreement

Participants generally disagree on the usefulness of group theory for the specific problem posed. While some acknowledge its broader applications, others maintain that it is not the right tool for the original inquiry.

Contextual Notes

The discussion highlights the ambiguity in the original question, which may affect the relevance of group theory. There are also references to alternative mathematical frameworks that could be more applicable, but no consensus is reached on the best approach.

Stephen Tashi
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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.
 
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Group theory? I don't think it's very useful. You should read up about elimination theory and Grobner bases though.
 
micromass said:
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)).
 
fresh_42 said:
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.
 

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