相关论文: On real and imaginary roots of generalised Okamoto…

We compute the precise leading asymptotics of the variance of the number of real roots for a large class of random polynomials, where the random coefficients have polynomial growth. Our results apply to many classical ensembles, including…

We present six Theorems on the univariate real Polynomial, using which we develop a new algorithm for deciding the existence of atleast one real root for univariate integer Polynomials. Our algorithm outputs that no positive real root…

A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational…

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…

Exact solutions of the dispersive and modified equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painleve equation, which arises as generalized scaling reductions of these equations.…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

We study the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlev\'e equation. Divisibility properties of the coefficients of these polynomials, concerning powers of 4,…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization…

This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number…

In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.

In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic…

We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, \dots, $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep)$ moments, and…

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…

We study dynamics of solutions in the initial value space of the sixth Painlev\'e equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general…

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends…

Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or…

This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for…