On the Number of Real Roots of a Random Algebraic Equation
نویسندگان
چکیده
1 . SOME time ago Littlewood and Offordt gave estimates of the number of real roots that an equation of degree n selected at random might be expected to have for various classes of equations in which the coefficients were selected on some probability basis . They found that, when each coefficient was treated on the same basis, the results were practically the same in all cases considered and agreed with those found for the family of equations fn(x) = 1+ElX+E2x2+. ..+En xn = 0 (1 .1) in which each E,,, v = 1, 2, . . ., n, is + 1 or -1 with equal probability . The object of this paper is to give a refinement of their result . We shall prove THEOREM . The number of real roots of most of the equations
منابع مشابه
An Lower Estimate for the Number of Level Crossing of a Random Algebraic Curve when the Coefficients follow Semi-Stable Distribution
Let ( ) ( ) ( ) ( ) be a sequence of mutually independent, identically distributed random variables following semi-stable distribution with characteristic function ( ( ) ) . In this work, we obtain the lower bound of the number of real zeros of the random algebraic equation∑ ( ) . ( ) denote the number of real roots must ( ⁄ ) , except for a set of measure at most .
متن کاملResearch Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation
where the aν(ω), ν= 0,1, . . . ,n, are random variables defined on a fixed probability space (Ω, ,Pr) assuming real values only. During the past 40–50 years, the majority of published researches on random algebraic polynomials has concerned the estimation of Nn(R,ω). Works by Littlewood and Offord [1], Samal [2], Evans [3], and Samal and Mishra [4–6] in the main concerned cases in which the ran...
متن کاملOn Classifications of Random Polynomials
Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...
متن کاملTopological Complexity and Schwarz Genus of General Real Polynomial Equation
We prove that the minimal number of branchings of arithmetic algorithms of approximate solution of the general real polynomial equation xd + a1xd−1 + · · · + ad−1x + ad = 0 of odd degree d grows to infinity at least as log2 d. The same estimate is true for the ε-genus of the real algebraic function associated with this equation, i.e. for the minimal number of open sets covering the space Rd of ...
متن کامل