Expected Number of Real Zeros of Gaussian Self-Reciprocal Random Algebraic Polynomials

2018 ◽  
Vol 42 (1) ◽  
pp. 105-113
Author(s):  
Soudabeh Shemehsavar
Keyword(s):  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials

scholarly journals Real zeros of random algebraic polynomials with binomial elements

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
A. Nezakati ◽  
K. Farahmand
Keyword(s):  
Algebraic Polynomial ◽  
Asymptotic Estimate ◽  
General Theorem ◽  
Random Variables ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials ◽  
Special Case

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0+a1x+a2x2+…+an−1xn−1. The coefficients aj(j=0,1,2,…,n−1) are assumed to be independent normal random variables with mean zero. For integers m and k=O(log⁡n)2 the variances of the coefficients are assumed to have nonidentical value var⁡(aj)=(k−1j−ik), where n=k⋅m and i=0,1,2,…,m−1. Previous results are mainly for identically distributed coefficients or when var⁡(aj)=(nj). We show that the latter is a special case of our general theorem.

scholarly journals Real zeros of classes of random algebraic polynomials

2003 ◽  
Vol 16 (3) ◽  
pp. 249-255 ◽  
Author(s):  
K. Farahmand ◽  
M. Sambandham
Keyword(s):  
Asymptotic Formula ◽  
Algebraic Polynomial ◽  
Random Coefficients ◽  
Asymptotic Estimates ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Algebraic Polynomials

There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn). This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj) the expected number of zeros of the polynomial increases to O(n). The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.

scholarly journals REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS

2008 ◽  
Vol 85 (1) ◽  
pp. 81-86 ◽  
Author(s):  
K. FARAHMAND
Keyword(s):  
Stable Distribution ◽  
Random Coefficients ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Symmetric Stable Distribution ◽  
Number Of Real Zeros ◽  
Significant Interest ◽  
Special Case

AbstractWe consider 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 $\theta _k={n\choose k}^{1/2}$. 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.

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