Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

Algebraic Polynomials with Non-identical Random Coefficients

The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial is known. The identical random coefficients a_j(ω) are normally distributed defined on a probability space , ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Q_n(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Q_n(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of .     

Complex Zeros of Algebraic Polynomial with Non-Zero Mean Random Coefficients

In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean and variance ². The value of the average density for the case of =0 and ²=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula.     

Algebraic polynomials with non-identical random coefficients

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial The coefficients are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all sufficiently large, the above expected value is shown to be . Also, it is known that if the have non-identical variance , then the expected number of real zeros increases to . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain . In fact, so far the case of is the only case that can significantly increase the expected number of real zeros.

     

Zeros of a random algebraic polynomial with coefficient means in geometric progression

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients {a_j}ⁿ⁻¹_j=0 of the polynomial T(x)=a₀+a₁x+a₂x²+?+a_n−1xⁿ⁻¹ are normally distributed, with mean E(a_j)=μ^j+1, where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of |μ|<1 and |μ|>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.     

The Expected Number of Local Maxima of a Random Algebraic Polynomial

In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial , where a _j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to log n, when n is large.     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

Exceedence Measure of Classes of Algebraic Polynomials

There is both mathematical and physical interest in the behaviour of the polynomial of the form . The coefficients a _j , j = 0,...,n are assumed to be independent normally distributed random variables with mean zero and variance ². In this paper by using the motion of exceedence measure for stochastic processes, for n large, we derive an asymptotic estimate for the expected area of the curve representing the above polynomial cut off by the x-axis. We show that our method can be used to obtain results for similar random polynomials.     

Number Variance of Random Zeros on Complex Manifolds

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set with smooth boundary is asymptotic to , where is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact K?hler manifold. Received: August 2006 Revision: March 2007 Accepted: April 2007     

The expected value of the number of real zeros of a random sum of Legendre polynomials

It is known that the expected number of zeros in the interval of the sum , in which is the normalized Legendre polynomial of degree and the coefficients are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to for large . We improve this result and show that this expected number is for any positive .

     

On the Roots of a Random System of Equations. The Theorem of Shub and Smale and Some Extensions

We give a new proof of a theorem of Shub and Smale on the expectation of the number of roots of a system of m random polynomial equations in m real variables, having a special isotropic Gaussian distribution. Further, we present a certain number of extensions, including the behavior as m → +∞ of the variance of the number of roots, when the system of equations is also stationary.     

Average Number of Real Roots of Random Harmonic Equations

Let {g_k}be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let _k (t ) (k = 0, 1, 2,...) be the normalized Jacobi polynomials orthogonal with respect to the interval [ – 1, 1 ]. Then it is proved that the average number of real roots of the random equations, _k=0 ⁿ g_kk(1)=C where Cis a constant, is asymptotically equal to n/in the same interval when nis large and even for C as long as C=O (n ²).     

On the Real Zeros of Positive Semidefinite Biquadratic Forms

For a positive semidefinite biquadratic forms F in (3, 3) variables, we prove that, if F has a finite number but at least 7 real zeros 𝒵(F), then it is not a sum of squares. We show also that if F has at least 11 zeros, then it has infinitely many real zeros and it is a sum of squares. It can be seen as the counterpart for biquadratic forms as the results of Choi, Lam, and Reznick for positive semidefinite ternary sextics.

We introduce and compute some of the numbers BB _{n, m} which are set to be equal to sup |𝒵(F)| where F ranges over all the positive semidefinite biquadratic forms F in (n, m) variables such that |𝒵(F)| < ∞.

We also recall some old constructions of positive semidefinite biquadratic forms which are not sums of squares and we give some new families of examples.     

On the Zeros of Orthogonal Polynomials: The Elliptic Case

Constructive Approximation - Let E = [–1, α] \cup [β, 1], –1 &;lt; α &;lt; β &;lt; 1, and let (pn) be orthogonal on E with respect to the weight function...