Complex Analogues of the Rolle's Theorem
2000 Mathematics Subject Classification: 30C10.Classical Rolle’s theorem and its analogues for complex algebraic polynomials are discussed. A complex Rolle’s theorem is conjectured.
Constructive function theory on sets of the complex plane through potential theory and geometric function theory.
Convex hulls of polyominoes.
Die Koeffizientenkörper schlichter Trinome.
Eine Anwendung des Gaußschen Integralsatzes.
Équations diophantiennes polynomiales à hautes multiplicités
On montre comment écrire de grandes familles, avec de hautes multiplicités, de cas d’égalité pour l’inégalité de Stothers-Mason (si sont des polynômes premiers entre eux, le nombre exact de racines du produit dépasse de le plus grand des degrés des composantes . On développera pour cela des techniques polynomiales itératives inspirées des décompositions de Dunford-Schwartz et de fonctions de Belyi. Des exemples d’application avec les conjectures ou de M. Hall sont développés.
Estimates for polynomials in the unit disk with varying constant terms
Let || · || be the uniform norm in the unit disk. We study the quantities Mn (α) := inf (||zP(z) + α|| - α) where the infimum is taken over all polynomials P of degree n - 1 with ||P(z)|| = 1 and α > 0. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that infα>0Mn (α) = 1/n. We find the exact values of Mn (α) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
Extention of Apolarity and Grace Theorem
MSC 2010: 30C10The classical notion of apolarity is defined for two algebraic polynomials of equal degree. The main property of two apolar polynomials p and q is the classical Grace theorem: Every circular domain containing all zeros of p contains at least one zero of q and vice versa. In this paper, the definition of apolarity is extended to polynomials of different degree and an extension of the Grace theorem is proved. This leads to simplification of the conditions of several well-known results...
Familles de polynômes ponctuellement bornées
Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle
We give a complete characterization of the positive trigonometric polynomials on the bi-circle, which can be factored as where is a polynomial nonzero for and . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...
Finite-infinite-range inequalities in the complex plane.
Functions without residue and a bilinear differential equation
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...
Growth of polynomials whose zeros are outside a circle
If p(z) be a polynomial of degree n, which does not vanish in |z| < k, k < 1, then it was conjectured by Aziz [Bull. Austral. Math. Soc. 35 (1987), 245-256] that [...] In this paper, we consider the case k < r < 1 and present a generalization as well as improvement of the above inequality.
Growth of polynomials with prescribed zeros
Heights of powers of Newman and Littlewood polynomials
Hénon mappings in the complex domain I : the global topology of dynamical space
Hyperbolic Fourth-R Quadratic Equation and Holomorphic Fourth-R Polynomials
MSC 2010: 30C10, 32A30, 30G35The algebra R(1; j; j2; j3), j4 = ¡1 of the fourth-R numbers, or in other words the algebra of the double-complex numbers C(1; j) and the corresponding functions, were studied in the papers of S. Dimiev and al. (see [1], [2], [3], [4]). The hyperbolic fourth-R numbers form other similar to C(1; j) algebra with zero divisors. In this note the square roots of hyperbolic fourth-R numbers and hyperbolic complex numbers are found. The quadratic equation with hyperbolic fourth-R...
Induced hyperbolicity for one-dimensional maps.
Inequalities concerning polar derivative of polynomials
In this paper we obtain certain results for the polar derivative of a polynomial [...] , having all its zeros on [...] which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.