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REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 85 / Issue 1 / August 2008
- Published online by Cambridge University Press:
- 01 August 2008, pp. 81-86
- Print publication:
- August 2008
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Weconsider a random algebraic polynomial of the form Pn,θ,α(t)=θ0ξ0+θ1ξ1t+⋯+θnξntn, where ξk, k=0,1,2,…,n have identical symmetric stable distribution with index α, 0<α≤2. First, for a general form of θk,α≡θk we derive the expected number of real zeros of Pn,θ,α(t). We then show that our results can be used for special choices of θk. In particular, we obtain the above expected number of zeros when
. The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of α on increasing the expected number of zeros compared with the special case of Gaussian coefficients.
. The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of Extrema of random algebraic polynomials with non-identically distributed normal coefficients
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 70 / Issue 2 / April 2001
- Published online by Cambridge University Press:
- 09 April 2009, pp. 225-234
- Print publication:
- April 2001
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An asymptotic estimate is derived for the expected number of extrema of a polynomial
whose independent normal coefficients possess non-equal non-zero mean values. A result is presented that generalizes in terms of normal processes the analytical device used for construction of similar asymptotic estimates for random polynomials with normal coefficients.
whose independent normal coefficients possess non-equal non-zero mean values. A result is presented that generalizes in terms of normal processes the analytical device used for construction of similar asymptotic estimates for random polynomials with normal coefficients.