FLQ, the Fastest Quadratic Complexity Bound on the Values of Positive Roots of Polynomials

In this paper we present F LQ, a quadratic complexity bound on the values of the positive roots of polynomials. This bound is an extension of FirstLambda, the corresponding linear complexity bound and, consequently, it is derived from Theorem 3 below. We have implemented FLQ in the Vincent-Akritas-Strzeboński Continued Fractions method (VAS-CF) for the isolation of real roots of polynomials and compared its behavior with that of the theoretically proven best bound, LM Q. Experimental results indicate that whereas F LQ runs on average faster (or quite faster) than LM Q, nonetheless the quality of the bounds computed by both is about the same; moreover, it was revealed that when VAS-CF is run on our benchmark polynomials using F LQ, LM Q and min(F LQ, LM Q) all three versions run equally well and, hence, it is inconclusive which one should be used in the VAS-CF method.

On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials

This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.

Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2 = β−1 = 0, p−1 = p−2 = 0, p0 (λ) = 1, p1 (λ) = cλ + b, c > 0, b ∈ C, λ ∈ C. It is shown that they are orthonormal on the real and the imaginary axes in the complex plane ...

Polynomials of Pellian Type and Continued Fractions

We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex- pansion of √fk (X) is related to that of √C, where √fk (X) = ak*X^2 +bk*X + C. This continues work in [1]–[4].

Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

∗ Research partially supported by INTAS grant 97-1644

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.

On Representations of Algebraic Polynomials by Superpositions of Plane Waves

* The author was supported by NSF Grant No. DMS 9706883.

On a Five-Diagonal Jacobi Matrices and Orthogonal Polynomials on Rays in the Complex Plane

∗ Partially supported by Grant MM-428/94 of MESC.

A Note on a Classical Generating Function for the Jacobi Polynomials

Mathematics Subject Classification: 33C45.

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45

On the Generalized Associated Legendre Functions

Mathematics Subject Classification: 33C60, 33C20, 44A15

Rolle's Theorem for Complex Polynomials with Real Coefficients

Let p(z) be an algebraic polynomial of degree n ¸ 2 with real coefficients and p(i) = p(¡i). According to Grace-Heawood Theorem, at least one zero of the derivative p0(z) is on the disk with center in the origin and radius cot(¼=n). In this paper is found the smallest domain containing at leas one zero of the derivative p0(z).

Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

AMS Subject Classification 2010: 11M26, 33C45, 42A38.

On a Stratification Defined by Real Roots of Polynomials

2000 Mathematics Subject Classification: 12D10

Overdetermined Strata in General Families of Polynomials

2000 Mathematics Subject Classification: 12D10.