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I am reading Gerald Follands Introduction To PDEs, and I got to this part:

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The part in red is a bit confusing, I am not sure how it follows from the sentences afterwards. I get that both are non-zero and homogenous, which means that we can somehow "standardize" the sum, but I am not sure how this follows for general alpha.

Laplacinator
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    Here's a toy model of the same computation. – Giuseppe Negro Dec 04 '17 at 12:03
  • @GiuseppeNegro Thank you! So are you saying that the expression (for simplicity assume the $a_{\alpha}$ to be constants)

    $$ |\xi|^k / \sum a_{\alpha} \xi^{\alpha} $$

    is homogenous of degree $0$, so that it is constant along rays, and so is determined by its maximum on the unit circle?

     – Laplacinator Dec 04 '17 at 12:40
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    The a_alpha ARE constant. The relevant variable here is $\xi$. So yes, that's what the book suggests to do – Giuseppe Negro Dec 04 '17 at 12:44
  • @GiuseppeNegro I see now, thank you! There is later a comment on this identity, that it is possible to show that there are $R,K$ such that for $\xi$ such that $|\xi| > R$ there is a $K$ such that $| \sum a_{\alpha} \xi^{\alpha} | \geq K|\xi|^m$ This feels like a trivial restatement of the first estimate, since it holds for all $\xi$. Am I missing something there? – Laplacinator Dec 04 '17 at 13:13
  • What is m? Surely something smaller than k, right? – Giuseppe Negro Dec 04 '17 at 13:17
  • @GiuseppeNegro Sorry, I meant $k$ of course.( I cant edit the post now) – Laplacinator Dec 04 '17 at 13:20
  • Yeah, I wouldn't overthink that. – Giuseppe Negro Dec 04 '17 at 13:32

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