Kac Formula

The expected number of Real zeros $E_n$ of a Random Polynomial of degree $n$ is

$$E_n = {1\over\pi}\int_{-\infty}^\infty \sqrt{{1\over(t^2-1)^2}-{(n+1)^2t^{2n}\over (t^{2n+2}-1)^2}}\,dt$$ (1)

$$= {4\over\pi}\int_0^1 \sqrt{{1\over(1-t^2)^2}-{(n+1)^2t^{2n}\over(1-t^{2n+2})^2}}\,dt.$$ (2)

As $n\to\infty$,

$$E_n={2\over\pi}\ln n+C_1 +{2\over \pi n}+{\mathcal O}(n^{-2}),$$ (3)

where

$$C_1={2\over\pi}\left[{\ln 2+\int_0^\infty \left({\sqrt{{1\ov... ...)^2}-{1\over x+1}}\,\,}\right)\,dx}\right]=0.6257358072\ldots.$$ (4)

The initial term was derived by Kac (1943).

References

Edelman, A. and Kostlan, E. ``How Many Zeros of a Random Polynomial are Real?'' Bull. Amer. Math. Soc. 32, 1-37, 1995.

Kac, M. ``On the Average Number of Real Roots of a Random Algebraic Equation.'' Bull. Amer. Math. Soc. 49, 314-320, 1943.

Kac, M. ``A Correction to `On the Average Number of Real Roots of a Random Algebraic Equation'.'' Bull. Amer. Math. Soc. 49, 938, 1943.