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This question pertains to harmonic analysis on spheres.

Let $H_d$ = {homogeneous, total degree $d$ harmonic polynomials in $\mathbb{C}[x_1,\dots,x_n]$}

Given that the

Dimension of $H_d = \binom {n+d-1}{ n-1} - \binom {n+d-3}{ n-1}$

How do I please show that the dimension of $H_d$ grows like $d^{n-2}$ as $d \rightarrow +\infty$

Thanks

2 Answers2

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The dimension of $H_d$ is a polynomial of degree at most $n-1$ in $d$. You need to show that the $d^{n-1}$ term vanishes and the $d^{n-2}$ term doesn't.

joriki
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  • @Andrew: What are $x$ and $y$? Your expression for $H_d$ only contains $n$ and $d$ as variables. – joriki Mar 03 '16 at 21:07
  • @Andrew: I see. Perhaps I misunderstood the question. I thought you were taking the expression for the dimension of $H_d$ as given and wanted to show that it grows as $d^{n-2}$. From your comment it seems you want to count harmonic polynomials -- presumably to derive the expression for the dimension of $H_d$? – joriki Mar 03 '16 at 21:18
  • @Andrew: I just noticed that I wrote "$H_d$" instead of "The dimension of $H_d$" in the answer -- I've corrected that. Perhaps this was what was causing the confusion? – joriki Mar 03 '16 at 21:20
  • Thanks, Joriki. The mistake and confusion are all mine. I'll start over. Previously I had put the binomial coefficients into factorial form and combined them over a common denominator. I'll go back and look at that with your original answer in mind. If I'm still stuck, maybe you wouldn't mind if I asked for further help. –  Mar 03 '16 at 21:33
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Well, $$\binom{n+d-1}{n-1} = \frac{(n+d-1)!}{(n-1)! d!}= \frac{(n+d-3)! (n+d-2)(n+d-1)}{(n-1)! (d-2)! (d-1)d} = \binom{n+d-3}{n-1} \frac{(n+d-2)(n+d-1)}{(d-1)d}.$$ So, your difference is $$\binom{n+d-3}{n-1}\left(\frac{(n+d-2)(n+d-1)}{(d-1)d} - 1\right).$$ The term in parentheses is asymptotic to $1/d,$ while the binomial coefficient is asymptotic to a polynomial in $d$ of degree $n-1.$

Igor Rivin
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