Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials

Charles N. Delzell

Displaying similar documents to “Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials”

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

The effect of rational maps on polynomial maps

Pierrette Cassou-Noguès (2001)

Annales Polonici Mathematici

Similarity:

We describe the polynomials P ∈ ℂ[x,y] such that $P (1 / v ⁿ, A ₁ v ⁿ + A ₂ v^{2 n} + . . . + A_{m - 1} v^{n (m - 1)} + v^{n m - k} w) \in ℂ [v, w]$ . As applications we give new examples of bad field generators and examples of families of polynomials with smooth and irreducible fibers.

Reduction and specialization of polynomials

Pierre Dèbes (2016)

Acta Arithmetica

Similarity:

We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is...

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

Similarity:

It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_{r}$ such that the polynomial $f (X) / N_{r}$ represents at least r distinct primes.

A formula for Jack polynomials of the second order

Francisco J. Caro-Lopera, José A. Díaz-García, Graciela González-Farías (2007)

Applicationes Mathematicae

Similarity:

This work solves the partial differential equation for Jack polynomials $C_{κ}^{α}$ of the second order. When the parameter α of the solution takes the values 1/2, 1 and 2 we get explicit formulas for the quaternionic, complex and real zonal polynomials of the second order, respectively.

Upper bounds for the $L_{q}$ norm of Fekete polynomials on subarcs

(2012)

Acta Arithmetica

Similarity:

Every compact set in $𝐂^{n}$ is a good compact set

Jan Erik Björk (1970)

Annales de l'institut Fourier

Similarity:

Let $K$ be an compact subset of an open set $V$ in $C^{n}$ . We show the existence of an open neighborhood $U$ of $K$ satisfying the following condition : if $f$ is holomorphic in $V$ and if there exists a sequence of polynomials which approximate $f$ uniformly in some open neighborhood $U_{f}$ of $K$ , there exists a sequence of polynomial which approximate $f$ uniformly in $U$ .

A note on the kernels of higher derivations

Jiantao Li, Xiankun Du (2013)

Czechoslovak Mathematical Journal

Similarity:

Let $k \subseteq k^{'}$ be a field extension. We give relations between the kernels of higher derivations on $k [X]$ and $k^{'} [X]$ , where $k [X] : = k [x_{1}, \dots, x_{n}]$ denotes the polynomial ring in $n$ variables over the field $k$ . More precisely, let $D = {D_{n}}_{n = 0}^{\infty}$ a higher $k$ -derivation on $k [X]$ and $D^{'} = {D_{n}^{'}}_{n = 0}^{\infty}$ a higher $k^{'}$ -derivation on $k^{'} [X]$ such that $D_{m}^{'} (x_{i}) = D_{m} (x_{i})$ for all $m \geq 0$ and $i = 1, 2, \dots, n$ . Then (1) $k {[X]}^{D} = k$ if and only if $k^{'} {[X]}^{D^{'}} = k^{'}$ ; (2) $k {[X]}^{D}$ is a finitely generated $k$ -algebra if and only if $k^{'} {[X]}^{D^{'}}$ is a finitely generated $k^{'}$ -algebra. Furthermore, we also show that the kernel $k {[X]}^{D}$ of a higher derivation $D$ of $k [X]$ can be generated...

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

In this paper we obtain certain results for the polar derivative of a polynomial $p (z) = c_{n} z^{n} + \sum_{j = μ}^{n} c_{n - j} z^{n - j}$ , $1 \leq μ \leq n$ , having all its zeros on $| z | = k$ , $k \leq 1$ , 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. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

Similarity:

Let p(z) be a polynomial of the form $p (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ , $a_{j} \in - 1, 1$ . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

The Daugavet equation for polynomials

Yun Sung Choi, Domingo García, Manuel Maestre, Miguel Martín (2007)

Studia Mathematica

Similarity:

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation $m a x_{| ω | = 1} | | I d + ω P | | = 1 + | | P | |$ for polynomials P: X → X. We show that this equation holds for every polynomial on the complex...

$0 - 1$ sequences having the same numbers of $(1 - 1)$ -couples of given distances

Antonín Lešanovský, Jan Rataj, Stanislav Hojek (1992)

Mathematica Bohemica

Similarity:

Let $a$ be a $0 - 1$ sequence with a finite number of terms equal to 1. The distance sequence $δ^{(a)}$ of $a$ is defined as a sequence of the numbers of $(1 - 1)$ -couples of given distances. The paper investigates such pairs of $0 - 1$ sequences $a, b$ that a is different from $b$ and $δ^{(a)} = δ^{(b)}$ .

Extendibility of polynomials and analytic functions on $ℓ_{p}$

Daniel Carando (2001)

Studia Mathematica

Similarity:

We prove that extendible 2-homogeneous polynomials on spaces with cotype 2 are integral. This allows us to find examples of approximable non-extendible polynomials on $ℓ_{p}$ (1 ≤ p < ∞ ) of any degree. We also exhibit non-nuclear extendible polynomials for 4 < p < ∞. We study the extendibility of analytic functions on Banach spaces and show the existence of functions of infinite radius of convergence whose coefficients are finite type polynomials but which fail to be extendible. ...