Dimension of a multivariate polynomial space

[dJesse]

Consider the space of all polynomials in n variables of degree at most d. The dimension of that space is C(n+d,d). How do I calculate the dimension of that same space when I restrict the domain of the polynomials to the unit ball? In that case all the polynomials (sum(i=1..n) x_i^2)^p with p a natural number are identical to the polynomial 1. One professor agrees with me that you have to subtract the cardinality of the set {(sum(i=1..n) x_i^2)^p | p in N} from C(n+d,d). But in my course text (written by another professor) sais that the correct formula is C(n+d,d)-C(n+d-2,d-2)

 

[rmehta]

I think the issue is that there are actually more polynomials that "vanish" when you restrict to the unit sphere (FYI, mathematicians use the word "ball" to include the interior of the sphere, where $\sum x^2 \leq 1$).

For example, if n=2 you know that x^2 + y^2 - 1 = 0 (on the unit sphere), but also, (x^2 + y^2 - 1)x = 0. In general, (x^2 + y^2 - 1)f(x) = 0 for any polynomial f(x). Since you're only looking at polynomials of degree d or less, you'll only want to f(x) to be of degree d-2 or less (so that (x^2 + y^2 -1)f(x) is of degree d or less). So the dimension of the space of "vanishing polynomials" is C(n+d-2,d-2), which is why you need to subtract this amount.

 

[kodylau]

hi, how to calculate the dimensionality of a polynomial space which has d variables and degree n? I googled but cannot find any answer. What kind of book should I read?

 

[chiro]

Just out of curiosity, does the mapping of x to f(x) have to be for all real numbers or just a particular subset?

 

[blue_raver22]

The dimension of a multivariate polynomial space can be calculated by using the formula C(n+d,d), where n is the number of variables and d is the maximum degree of the polynomials in that space. However, when the domain is restricted to the unit ball, the dimension of the polynomial space changes. This is because all polynomials of the form (sum(i=1..n) x_i^2)^p, where p is a natural number, are identical to the polynomial 1. This means that the dimension of the polynomial space is reduced, as all these polynomials can be represented by a single polynomial.

One way to calculate the dimension of the polynomial space in this case is to subtract the cardinality of the set {(sum(i=1..n) x_i^2)^p | p in N} from the original formula C(n+d,d). This is because the set contains all the redundant polynomials that can be represented by the polynomial 1. This approach is supported by one professor.

However, the course text written by another professor suggests a different formula, C(n+d,d)-C(n+d-2,d-2). This formula also takes into account the reduction in dimension due to the restriction of the domain to the unit ball. It is possible that this formula is based on a different approach or perspective.

In any case, it is important to understand the reasoning and assumptions behind each formula in order to accurately calculate the dimension of the polynomial space in a given scenario. Both formulas may be valid in different contexts and it is up to the individual to determine which one is appropriate for their specific problem. It would be beneficial to consult with both professors and try to understand their perspectives in order to gain a better understanding of the concept.