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Is the largest root of a random real polynomial more likely real than complex?
This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO. The number of real roots of a ra...
And this probability decreases to some value near$1/2$ as$n \to \infty$ as shown in the above graph (created using a Monte Carlo simulation with$10^5$ trials for each value of$n$). Question 2: Does the probability that the largest (or the smallest) root of a polynomial of degree $n$ is real converge (to some value near $\frac{1}{2}$ as $n \to \infty$)? Update: In the linked MSE post, it has now been proved that the probability that the largest root is real is at least
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