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) / π.
A higher order asymptotic series is
2 log(n) / π + 0.6257… + 2/nπ + 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.
Related posts
[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
Sigh. Must be getting cabin fever due to too much time off.
The first thought entering my mind after reading the title “Expected number of roots” was “after pulling too fine a comb through your hair.”
Sorry-not-sorry.
interesting
What about the expected number of real roots of a polynomial with real coefficients drawn randomly from the entire set of reals? (Would that be a uniform distribution of all real numbers?)