Polynomials in n variables subspaces and subrepresentations

In summary: And how do polynomials represent vector spaces? Also, what is the significance of the degree of a polynomial in relation to the dimensions of the vector space? Additionally, the discussion later on involves decomposing a polynomial of a certain degree into subspaces based on partitions. Can someone explain this concept further?
  • #1
PsychonautQQ
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Homework Statement


Trying to make sense of my notes...
"A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form:
p(x)= ∑(C_i)x^β"

so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is how many dimensions the vector space is? I'm quite confused.

Later it talks about decomposing a polynomial of degree k into subspaces spanned by monomials of a particular "type" that are labelled by partitions of k. Example:
(x^2)(y^2)z and x(y^2)(w^2) are both in P_(2,2,1,0)(x,y,z,w)

anyone have any idea what any of this means?



Homework Equations





The Attempt at a Solution

 
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  • #2
PsychonautQQ said:

Homework Statement


Trying to make sense of my notes...
"A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form:
p(x)= ∑(C_i)x^β"

so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is how many dimensions the vector space is? I'm quite confused.

Later it talks about decomposing a polynomial of degree k into subspaces spanned by monomials of a particular "type" that are labelled by partitions of k. Example:
(x^2)(y^2)z and x(y^2)(w^2) are both in P_(2,2,1,0)(x,y,z,w)

anyone have any idea what any of this means?



Homework Equations





The Attempt at a Solution


What is the definition of a vector space?
 

1. What are polynomials in n variables subspaces and subrepresentations?

Polynomials in n variables subspaces and subrepresentations are mathematical objects that consist of expressions made up of variables and coefficients. These expressions can be added, subtracted, and multiplied together to form new expressions. They are often used in algebra, calculus, and other branches of mathematics to model real-world problems.

2. How are polynomials in n variables subspaces and subrepresentations different from regular polynomials?

Polynomials in n variables subspaces and subrepresentations differ from regular polynomials in that they involve multiple variables, rather than just one or two. This allows for a more complex and versatile representation of mathematical concepts and problems.

3. What is the importance of studying polynomials in n variables subspaces and subrepresentations?

Studying polynomials in n variables subspaces and subrepresentations can help us better understand and solve a wide range of mathematical problems. These concepts are also important in many areas of science and engineering, such as physics, computer science, and statistics.

4. How do you find the degree of a polynomial in n variables subspace or subrepresentation?

The degree of a polynomial in n variables subspace or subrepresentation is determined by the highest exponent of any variable in the expression. For example, in the polynomial 2x^2y^3z, the degree would be 3.

5. Can polynomials in n variables subspaces and subrepresentations be graphed?

Yes, polynomials in n variables subspaces and subrepresentations can be graphed, but the resulting graph may not be a traditional curve like in regular polynomials. Instead, it may be a higher-dimensional shape or surface. The graph can still be used to visualize and analyze the behavior of the polynomial and its solutions.

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