相关论文: An iterative approach for real roots of polynomial…

We study the real roots of the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlev\'e equation. It has been conjectured that the number of real roots of the nth…

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…

In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…

An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this…

The purpose of this article is to present, in a simple way, an analytic approach to special numbers and polynomials. The approach is based on the derivative polynomials. The paper is, to some extent, a review article, although it contains…

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating…

Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the…

For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities…

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a…

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…

In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…

An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative…

Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the B\'ezout number. A similar result is known for random multi-homogeneous…

We describe new algorithms to compute Whitney stratifications of real algebraic varieties. Using either conormal or polar techniques, these algorithms stratify a complexification of a given real variety. We then show that the resulting…

We give an explicit formula for the correlation functions of real zeros of a random polynomial with arbitrary independent continuously distributed coefficients.

The notion of root polynomials of a polynomial matrix $P(\lambda)$ was thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020]. In this…

We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.

Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance…