# Applying group theory to multivariate eqs

• A
• Stephen Tashi
In summary, the conversation revolves around the usefulness of group theory in solving multivariate algebraic equations. Some examples of applications of group theory in various fields are mentioned, but it is also acknowledged that it may not be useful in all cases. The conversation also touches upon the vagueness of the original question and how group theory can be helpful in certain cases.
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.

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.

## 1. What is group theory?

Group theory is a branch of mathematics that studies the properties and structures of groups, which are mathematical objects that consist of a set of elements and a binary operation that combines any two elements to form a third element. It has many applications in various fields, including physics, chemistry, and computer science.

## 2. How is group theory applied to multivariate equations?

In the context of multivariate equations, group theory is used to identify symmetries and patterns in the equations and their solutions. These symmetries can then be used to simplify the equations, making it easier to find solutions or understand their properties.

## 3. What are the advantages of using group theory in solving multivariate equations?

One of the main advantages of using group theory in solving multivariate equations is that it allows for the identification of hidden relationships and patterns that may not be immediately obvious. This can lead to more efficient and elegant solutions, as well as a deeper understanding of the equations and their solutions.

## 4. Are there any limitations to applying group theory to multivariate equations?

While group theory can be a powerful tool in solving multivariate equations, it does have limitations. It may not be applicable to all types of equations, and in some cases, the symmetries and patterns identified may not lead to a simpler solution. Additionally, the use of group theory requires a strong understanding of both group theory and the specific equations being studied.

## 5. How can group theory be used to solve real-world problems?

Group theory has numerous applications in the real world, including in physics, chemistry, and computer science. For example, it can be used to analyze the symmetries of molecules and crystals, which can help in predicting their physical and chemical properties. It can also be used in cryptography to create secure encryption algorithms that rely on the symmetries of groups. Overall, group theory allows for a deeper understanding of complex systems and can aid in finding efficient and elegant solutions to real-world problems.

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