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Kac Formula


KacFormula

The expected number of real zeros E_n of a random polynomial of degree n if the coefficients are independent and distributed normally is given by

E_n=1/piint_(-infty)^inftysqrt(1/((t^2-1)^2)-((n+1)^2t^(2n))/((t^(2n+2)-1)^2))dt
(1)
=4/piint_0^1sqrt(1/((1-t^2)^2)-((n+1)^2t^(2n))/((1-t^(2n+2))^2))dt.
(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
(3)

(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand I_n(t) (left) and numerical values of E_n (red curve in right plot) for small n. The first few values are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955, ....

As n->infty,

 E_n=2/pilnn+C_1+2/(pin)+O(n^(-2)),
(4)

where

C_1=2/pi{ln2+int_0^infty[sqrt(1/(x^2)-(4e^(-2x))/((1-e^(-2x))^2))-1/(x+1)]dx}
(5)
=0.6257358072...
(6)

(OEIS A093601; top curve in right plot above). The initial term was derived by Kac (1943).


See also

Random Polynomial

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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."

Referenced on Wolfram|Alpha

Kac Formula

Cite this as:

Weisstein, Eric W. "Kac Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KacFormula.html

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