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Let $n$ be an integer greater than 1. How must $n$ numbers $a_i$ in the interval $[0,1]$ be chosen that the vandermonde polynomial $$\prod_{i\ne j} (a_i-a_j) $$ has the maximum possible value ?

My conjecture, that the numbers must be equidistant, seems to be false due to numerical analysis.

martini
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Peter
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  • Yes I did, I realized why equidistant is false. (WLOG $a_i$ are in decreasing order) Note that if we increase $a_1$ to 1 and decrease $a_n$ to 0, we're only increasing each individual term (or stay constant), hence the product will increase. – Calvin Lin Jun 19 '13 at 14:44
  • If the product is taken symmetrically then the maximum is zero as it can never be strictly positive. If the product is only taken for $i<j$ the result might be different. – S.B. Jun 19 '13 at 14:55
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    See http://math.stackexchange.com/q/15366 – ˈjuː.zɚ79365 Jun 19 '13 at 14:57
  • OK, I seems that this definition of the vandermonde polynomial is false because the value must be positive or 0. With i<j and (a(i)-a(j))^2 , the definition should be OK. – Peter Jun 19 '13 at 15:03
  • I really did not know that this question was already asked. Sorry. – Peter Jun 19 '13 at 15:05

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