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Suppose I have a polynomial ring $k[x_0,\cdots,x_n]$ and let $k[x_0,\cdots,x_n]_m$ denote the homogeneous part with degree $m$. How can I see that this $k[x_0,\cdots,x_n]_m$ has dimension $\binom{m+n}{m}$? Or more exactly, why does this give the Krull dimension?

Christina
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    The monomials are linearly independent, to count them https://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition – reuns Nov 17 '20 at 20:41
  • But is that same as the Krull dimension? – Christina Nov 17 '20 at 23:08
  • No. The Krull dimension $=n$ is the idea that we have a chain of $n+1$ prime ideals : ${0}\subset (x_0)\subset (x_0,x_1)\subset \ldots \subset (x_0,\ldots,x_n)$. The homogeneous polynomials of degree $m$ are those of the form $f(x)=\sum_{|a|=m} c_a x^a$ (where $x^a= \prod_j x_j^{a_j}$). Note that $f(x)/x_0^m$ is well-defined as a function on $\Bbb{P}^n$. – reuns Nov 17 '20 at 23:19

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Hint:

If you denote $k_i$ the exponent of $xi$in such a monomial, it is the number of solutions in natural numbers of the equation $$k_0+k_1+\dots+ k_n=m.$$

Bernard
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