How to calculate the $k$-dimension of a subspace of a polynomial ring?

Asked Jul 23 '15 at 11:00

Active Jul 23 '15 at 11:00

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Let $k$ be an infinite field and $R:=k[x_1,...,x_n]$ the polynomial ring in $n$ indeterminates.

Why is the $k$-dimension of $U$ given by $\begin{pmatrix} n+m-1 \\ m\end{pmatrix}$, when $U$ is the subspace of $R$ defined by the $k$-span of those polynomials which are homogeneous of degree $m$ ?

Thanks for the help!

asked Jul 23 '15 at 11:00

LSt

2

Hint: How many $n$-tuples $\left(r_1,r_2,\ldots,r_n\right)$ of nonnegative integers are there such that $r_1+r_2+\ldots+r_n=m$? – Batominovski Jul 23 '15 at 11:05
1

It's a stars-and-bars computation. – hardmath Jul 23 '15 at 15:45

How to calculate the $k$-dimension of a subspace of a polynomial ring?

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