Polynomials with integer coefficients and their zeros

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.     

Finite groups whose character degree graphs coincide with their prime graphs

In the literature, there are several graphs related to a finite group G. Two of them are the character degree graph, denoted by ΔG), and the prime graph ΓG), In this paper we classify all finite groups whose character degree graphs are disconnected and coincide with their prime graphs. As a corollary, we find all finite groups whose character degree graphs are square and coincide with their prime graphs.     

Polynomials dertermined by a few of their coefficients

We prove some results which indicate that a monic polynomial over a field of characteristic zero with exactly κ distinct zeros may be determined up to finitely many possibilities by any κ of its nonzero proper coefficients.     

Polynomials with all zeros on the unit circle

We give a new suffcient condition for all zeros of self-inversive polynomials to be on the unit circle, and find the location of zeros. This generalizes some recent results of Lakatos [7], Schinzel [17], Lakatos and Losonczi [9], [10]. By this suffcient condition the mentioned results can be treated in a unified way.     

Polynomials with prescribed zeros on an analytic curve

Recently Totik and Varjú [17] presented an estimate of the uniform norm of a monic polynomial with prescribed zeros on the unit circle. In this paper we improve their estimate and extend it to the case of polynomials with some zeros on an arbitrary analytic curve in the complex plane.     

Polynomials with nonnegative coefficients and a given factor

For a given real polynomial f without positive roots (without nonnegative roots, respectively) we study polynomials h of lowest degree such that the product hf has nonnegative (positive, respectively) coefficients. Our main result is the determination of polynomials h of minimal degree with the aforementioned properties for quadratic polynomials f with negative discriminant; here we exploit slightly modified ideas of E. Meissner (1911). Further, we show that h can be calculated in finitely many steps.     

On the zeros of recurrence sequences with non-constant coefficients

The paper deals with the zeros of sequences, which satisfy linear recurrences with non-constant coefficients. We prove a weaker analogon of the Skolem-Mahler-Lech theorem on ordinary recurrent sequences. The proof relies on Szemerédi's theorem on arithmetic progressions in sets of positive density.     

Some properties of zeros of polynomials with vanishing coefficients

It is well-known that when a polynomial whose coefficients are continuous functions of a parameter loses its degree then some of its zeros must vanish at infinity. In this paper, we consider such a situation: we examine how roots of a complex polynomial tend to infinity as some of its coefficients, including the leading one, tend to zero. We show, among other things, that in such a situation the unbounded paths traced by the roots of the polynomial have asymptotes; we also obtain their formulas. Some examples are presented to complete and illustrate the results.     

On the coefficients of a polynomial with restricted zeros

LetP(Z)=αn Zn + αn-1Zn-1 +…+α0 be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coeffi-cients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).     

Sequences of polynomials whose zeros lie on fixed lemniscates

For fixed integersk2 we study sequences of polynomialsP _n (z) with the following properties: (i) degP _n ; (ii) the zeros of all theP _n (z) lie on a certain lemniscate withk ₁k foci, one of which is the origin; (iii) theP _n (z) can be cut in such a way that the zeros of the lower part all lie on the unit circle and those of the upper part lie on a lemniscate having the foci in (ii) excluding the origin. Several special cases and examples are considered.