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I am interested in determining the most probable maximum values of the real roots of polynomials of form $P(x)=\sum_{k=0}^n a_{k} x^k$ where the degree $n$ will have a defined value (say 3,4,5...) and $a_k$ are chosen from the set $\{-1,1\}$ with equal probability. Previously I had considered the maximum possible root for a set degree (see my previous question).

So in this case, I'm more looking to:

  1. Enumerate all possible polynomials for a given degree $n$ (e.g. Tuples[{-1, 1}, {4}].{1, x, x^2, x^3} for $n=3$)

  2. For each possible enumerated polynomial $P(x)$ for a given degree, find the maximum $|root|$.

  3. Plot a (bar?) graph of the maximum $|root|$ values and their occurrences. (For a set degree)

So, in this case, rather than finding the maximum possible root a polynomial could have for a set degree $n$, I'd like to graphically see the most probable largest roots of $P(x)$ for $n$ fixed.

Any suggestions/help with this issue is immensely appreciated. Thank You!

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  • $\begingroup$ You are aware, I presume, that the number of polynomials of degree n that you describe grows as 2^n. The problem very quickly becomes intractable. $\endgroup$ – m_goldberg Jul 23 '16 at 20:34
  • $\begingroup$ @m_goldberg I am aware, I have a supercomputer at my disposal, and only wish to consider some lower degrees. For higher degrees, as I've considered different aspects of this problem before, I simply do a sample of polynomials instead of using Tuples to do every single possible polynomial. But yes, this problem very quickly becomes intractable! $\endgroup$ – David_Shmij Jul 23 '16 at 20:38
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Well, I managed to get the job done in the following way:

roots = x /. NSolve[# == 0, x, Reals] & /@ (Tuples[{-1, 1}, {4}].{1, x, x^2, 
 x^3}) 
absroots = Abs[roots]
toplot = Map[Max, absroots, 1]
Histogram[toplot]
toplot2 = Tally[toplot]
ListPlot[toplot2]

It may not be the cleanest/easiest way, but with simply altering the first line I can do many degrees of $n$.

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