Kac Formula
المؤلف:
Edelman, A. and Kostlan, E.
المصدر:
"How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32
الجزء والصفحة:
...
18-3-2021
3451
Kac Formula
The expected number of real zeros 
of a random polynomial of degree 
if the coefficients are independent and distributed normally is given by
(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.gif) |
(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 are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955, ....
As 
,
 |
(4)
|
where
(OEIS A093601; top curve in right plot above). 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.
Kostan, E. "On the Distribution of Roots in a Random Polynomial." 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.
Sloane, N. J. A. Sequence A093601 in "The On-Line Encyclopedia of Integer Sequences."
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