On Roots of Polynomials and of their Derivatives

2015

In this paper we investigate the problem of simultaneous factorization of second degree polynomials with positive integer coefficients.

Pacific Journal of Mathematics, 1982

A general theorem concerning the structure of a certain real algebraic curve is proved. Consequences of this theorem include major extensions of classical theorems of Pόlya and Schur on the reality of roots of polynomials.

Journal of Nonlinear Mathematical Physics, 2017

We evaluate the number of monic polynomials (of arbitrary degree N) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property peculiar polynomials. We further show that the problem of determining the peculiar polynomials of degree N simplifies when any of the coefficients is either 0 or 1. We proceed to estimate the numbers of peculiar polynomials of degree N having one coefficient zero, or one coefficient equal to one, or neither.

Publications de l'Institut Math?matique (Belgrade)

We extend Aziz and Mohammad's result that the zeros, of a polynomial P (z) = n j=0 a j z j , ta j a j−1 > 0, j = 2, 3,. .. , n for certain t (> 0), with moduli greater than t(n − 1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P (z), of degree n, with complex coefficients, does not vanish in the disc |z − ae iα | < a/(2n); a > 0, max |z|=a |P (z)| = |P (ae iα)|, for r < a < 2, r being the greatest positive root of the equation x n − 2x n−1 + 1 = 0, and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).

2012

In this paper, we establish some relations between the zeros and coefficients of a polynomial and thereby prove a few results concerning stable polynomials.

In this paper, we consider the zero distributions of q-shift&nbsp;monomi-als&nbsp;and difference polynomials of meromorphic functions with zero order,&nbsp;that extends the classical Hayman results on the zeros of differential&nbsp;poly-nomials&nbsp;to q-shift difference polynomials. We also investigate&nbsp;problem&nbsp;of&nbsp;q-shift difference polynomials that share a common value.

Publications de l'Institut Mathematique, 2011

Let α be an algebraic number with no nonnegative conjugates over the field of the rationals. Settling a recent conjecture of Kuba, Dubickas proved that the number α is a root of a polynomial, say P , with positive rational coefficients. We give in this note an upper bound for the degree of P in terms of the discriminant, the degree and the Mahler measure of α; this answers a question of Dubickas.

Illinois Journal of Mathematics, 1997

Journal of Classical Analysis, 2014

We establish necessary and sufficient conditions for an arbitrary polynomial of degree n, especially with only real roots, to be trivial, i.e. to have the form a(x − b) n. To do this, we derive new properties of polynomials and their roots. In particular, it concerns new bounds and genetic sum representations of the Abel-Goncharov interpolation polynomials. Moreover, we prove the Sz.-Nagy type identities, the Laguerre and Obreshkov-Chebotarev type inequalities for roots of polynomials and their derivatives. As applications these results are associated with the known problem, conjectured by Casas-Alvero in 2001, which says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. We investigate particular cases of the problem, when the conjecture holds true or, possibly, is false.

2012

Let P(z) be a polynomial of degree n with real or complex coefficients . The aim of this paper is to obtain a ring shaped region containing all the zeros of P(z). Our results not only generalize some known results but also a variety of interesting results can be deduced from them.

2015

In this paper we consider the problem of finding the number of zeros of a polynomial in a prescribed region by subjecting the real and imaginary parts of its coefficients to certain restrictions.

2012

We study the following problem raised by von zur Gathen and Roche [GR97]:

1994

General Mathematics, 2005

It is well-known that, if f ∈ R[X], deg(f) = n ≥ 2 and Df divides f , then f is a scalar multiple of the n-th power of a monic polynomial of first degree, X +a, with a certain a ∈ R (it can be proved solving a simple differential equation which contains the associated polynomial function of f and its derivative). The converse assertion is obvious. In this paper, in the main result, we will show that, adding a simple supplimentar normating condition, the two classes defined by the mentioned properties also coincide with the class of the polynomials f which are reciprocal simultaneously with Df ; but it results that a = 1. This result also will be considered in the general situation of the polynomials of K[X], where K is an infinite commutative field an we will use only the formal derivative D. Finally we will pass in the umbral calculus and we will transpose the result in the case of a certain delta operator Q, in relation to its basic sequence (p n) n .

A high school content on polynomial functions. Prepared to aid students as well as instructors.