Polynomials in n variables subspaces and subrepresentations

[PsychonautQQ]

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

 

[Ray Vickson]

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?