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For which pairs of integers $(n,m)$ is the maximum of the following function $$f(x)=\sum_{i_1+\dots +i_n=m}\prod_{k=1}^n x^{i_k}_{k},\ \ x=(x_1,\dots,x_n), \|x\|=1$$ attained when $x_1=\dots=x_n$?

(Note: the OP has noted that this question arises from research into Bergman projections, and assures us it is far from trivial -- Todd Trimble.)

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    $\begingroup$ FYI, $f(x) = \sum_{k=1}^n x_k^{n+m-1} \cdot \prod_{\ell \ne k}(x_k - x_\ell)^{-1}$ when no two components of $x$ are equal. $\endgroup$
    – Steve Huntsman
    Commented Aug 13, 2013 at 14:15
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    $\begingroup$ Hi user36162: The down votes are probably in response to the question being entirely unmotivated. Background, what you've tried, and motivation are very important at MathOverflow. Why is this an interesting research question? Some further discussion of what makes a good MO question is at meta.mathoverflow.net/questions/70/how-to-ask-page. When you edit the question to include this part, it will automatically get nominated for reopening, and we'll look it over again. $\endgroup$
    – Theo Johnson-Freyd
    Commented Aug 13, 2013 at 14:43
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    $\begingroup$ Just to be clear: I was not one of the downvoters, and I half-expected the problem was in fact difficult and not at all homework -- my questions were meant to draw out user36162 a bit. I think it's the elementary formulation which makes it a bit hard to evaluate as to whether it's "research level", but I'm inclined to take OP at his/her word and vote to reopen, with a word of advice to OP that it's especially questions that look possibly elementary and are unmotivated that are susceptible to being shut down. Hope he/she won't take this to heart. Voting to reopen. $\endgroup$
    – Todd Trimble
    Commented Aug 13, 2013 at 22:21
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    $\begingroup$ @ToddTrimble: I will support a reopen vote just as soon as OP adds some background and motivation. I ask for various reasons. One is that I learn quite a lot from reading MO questions, provided they include such material. $\endgroup$
    – Theo Johnson-Freyd
    Commented Aug 13, 2013 at 22:50
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    $\begingroup$ 'I think it's the elementary formulation which makes it a bit hard to evaluate as to whether it's "research level"'. Can you solve it in under 1 hour? If not, and if you are an "established researcher", it certainly is. It MAY be trivial after all but sheer politeness requires that you convince yourself about it first before trying to convince anyone else. Voting to reopen for now. :) Of course, the comments about background and motivation stand but the mere absence of those may be grounds for downvoting but not for the closure, IMHO. $\endgroup$
    – fedja
    Commented Aug 13, 2013 at 23:16

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