Real zeros of a random polynomial with Legendre elements
نویسندگان
چکیده
منابع مشابه
The Real Zeros of a Random Polynomial with Dependent Coefficients
Abstract. Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of 2 π logn, as n → ∞. Several years later, Sambandham considered two cases with some dependence assumed among the coefficients. The first case ...
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Consider the random hyperbolic polynomial, f(x) = 1a1 coshx+···+np × an coshnx, in which n and p are integers such that n ≥ 2, p ≥ 0, and the coefficients ak(k = 1,2, . . . ,n) are independent, standard normally distributed random variables. If νnp is the mean number of real zeros of f(x), then we prove that νnp = π−1 logn+ O{(logn)1/2}.
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Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, . . . , tn) projected onto the surface of the unit sphere, divided by π. The probability density...
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ژورنال
عنوان ژورنال: Journal of Applied Mathematics and Stochastic Analysis
سال: 1997
ISSN: 1048-9533,1687-2177
DOI: 10.1155/s1048953397000324