We study p-adic root separation for quadratic and cubic polynomials with integer coefficients. The quadratic and reducible cubic polynomials are completely understood, while in the irreducible cubic case, we give a family of polynomials with the bound which is the best currently known.

Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph)? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomials?

A family of the Apostol-type polynomials was introduced and investigated recently by Luo and Srivastava (see (Appl. Math. Comput. 217:5702-5728, 2011)). In this paper, we study this polynomial family on P, the algebra of polynomials in a single variable x over all linear functional on P. By using the way of the umbral algebra, we obtain some fundamental properties of the generalized Apostol-typ...

We prove generalized arithmetic-geometric mean inequalities for quasi-means arising from symmetric polynomials. The inequalities are satisfied by all positive, homogeneous symmetric polynomials, as well as a certain family of nonhomogeneous polynomials; this family allows us to prove the following combinatorial result for marked square grids. Suppose that the cells of a n × n checkerboard are e...

The paper considers the robust stability veriication of polynomials with polynomial parameter dependency. A new algorithm is presented which relies on the expansion of a multivariate polynomial into Bernstein polynomials and is based on the inspection of the value set of the family of polynomials on the imaginary axis. It is shown how an initial interval on the imaginary axis through which zero...

We give a family of D5-polynomials with integer coefficients whose splitting fields over Q are unramified cyclic quintic extensions of quadratic fields. Our polynomials are constructed by using Fibonacci, Lucas numbers and units of certain cyclic quartic fields.

We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras quantized enveloping algebras Kac-Moody type. Our methods are based on star products noncommutative $\mathbb{N}$-graded algebras. The resulting expressed in terms continuous q-Hermite polynomials and a new family deformed Chebyshev polynomials.

Bezier curves (BC) are important tools in a wide range of diverse and challenging applications, from computer aided design to generic object shape descriptors. A major constraint of the classical BC is that only global information concerning control points (CP) is considered, consequently there may be a sizeable gap between the BC and its control polygon (CtrlPoly), leading to a large distortio...

It is unusual for an irreducible polynomial to have a root with rational real part or with rational imaginary part. Of course, such polynomials exist: one can simply take the minimal polynomial of, say, 1+ i √ 2 or √ 2+ i. The same applies to polynomials having a root of rational modulus. But it turns out to be of interest to characterize these three kinds of polynomials. We therefore define ou...