TOPICS
Kac Formula
The Kac formula gives the expected number 
of real roots of a random
polynomial of degree 
whose coefficients are
independent, each with the standard normal distribution:
 |  |  |
(1)
|
 |  |  |
(2)
|
(Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by
![E_n=1/piint_(-infty)^inftysqrt([(partial^2)/(partialxpartialy)ln(1-(xy)^(n+1))/(1-xy)]_(x=y=t))dt](https://mathworld.wolfram.com/images/equations/KacFormula/NumberedEquation1.svg) |
(3)
|
(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand 
(left) and numerical values of
(red curve in right plot) for small
. The first few values, for 
, 2, ..., are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955,
....
As 
,
 |
(4)
|
where
 |  | ![2/pi{ln2+int_0^infty[sqrt(1/(x^2)-(4e^(-2x))/((1-e^(-2x))^2))-1/(x+1)]dx}](https://mathworld.wolfram.com/images/equations/KacFormula/Inline16.svg) |
(5)
|
 |  |  |
(6)
|
(OEIS A093601; top curve in right plot above). The leading asymptotic term was derived by Kac (1943).
See also
Random PolynomialExplore with Wolfram|Alpha
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.Kostlan, E. "On the Distribution of Roots of Random Polynomials." Ch. 38 in From Topology to Computation: Proceedings of the Smalefest (Ed. M. W. Hirsch, J. E. Marsden, and M. Shub). New York: Springer-Verlag, pp. 419-431, 1993. https://doi.org/10.1007/978-1-4612-2740-3_38.Sloane, N. J. A. Sequence A093601 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Kac FormulaCite this as:
Weisstein, Eric W. "Kac Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KacFormula.html