Expected number of roots

Suppose you create a 100th degree polynomial by picking coefficients at random from a standard normal. How many real roots would you expect?

There are 100 complex roots by the fundamental theorem of algebra, but how many would you expect to be real?

A lot fewer than I would have expected. There's a theorem due to Kac [1] that the expected number of zeros is asymptotically

2 log(n) / .

A higher order asymptotic series is

2 log(n) / + 0.6257... + 2/n + O(n^-2)

So for an 100th degree polynomial, you'd expect around 3 or 4 real roots.The number of roots had to be even-the only way to have an odd number of roots is with multiple roots, and such roots have probability zero-and so four roots would be common. Of course there could be up to 100 real roots, but that is extremely unlikely.

It turns out a polynomial would have to have degree over two million before the expected number of roots would 10.

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[1] Alan Edelman and Eric Kostlan. How many zeros of a random polynomial are real? Bulletin of the AMS. Volume 32, Number 1, January 1995

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