Let $V_m$ be the vector space of polynomials in $k[x_1,...,x_n]$ with degree at most $m$. Then what is the dimension of this vector space?

Question

Let $V_m$ be the vector space of polynomials in $k[x_1,...,x_n]$ with degree at most $m$. Then what is the dimension of this vector space?

I can do this with m<4 maybe. But what is the general formula? It seems that this is a combinatorics problem. But I really do not have idea how to find a formula.

@AloizioMacedo I am so sorry. I typed a wrong title. So the polynomial should be in $k[x_1,...,x_n]$. This is more complicated I guess. — , Oct 14 '15 at 04:01
I have voted to reopen this question, since it does not really appear to be a duplicate of the other question. Note that the given question asks for polynomials of degree at most $m$, while the other question considers polynomials of degree precisely $m$. The difference is not too big, but still... — PhoemueX, Nov 12 '19 at 10:40

score 6 · Accepted Answer · edited Oct 16 '15 at 20:02

6

The relevant formulas appear for example in Wendland's "Scattered Data Approximation", part 2 "Haar spaces and multivariate polynomials", Theorem 2.5. In their notation proves for the dimension of $d$-variate polynomials of degree $m$ that $\dim \pi_m(\mathbb{R}^d)={m+d\choose d}$.

edited Oct 16 '15 at 20:02

user26857

52,094

answered Oct 16 '15 at 03:24

rych

4,205

Let $V_m$ be the vector space of polynomials in $k[x_1,...,x_n]$ with degree at most $m$. Then what is the dimension of this vector space?

1 Answers1