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

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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.

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p, n)$ , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials. ...

The effect of rational maps on polynomial maps

Pierrette Cassou-Noguès (2001)

Annales Polonici Mathematici

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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

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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...

Symmetric identity for polynomial sequences satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$

Farid Bencherif, Rachid Boumahdi, Tarek Garici (2021)

Communications in Mathematics

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Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$ with $A_{0} (x)$ a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

A generalisation of Amitsur's A-polynomials

Adam Owen, Susanne Pumplün (2021)

Communications in Mathematics

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We find examples of polynomials $f \in D [t; σ, δ]$ whose eigenring $ℰ (f)$ is a central simple algebra over the field $F = C \cap Fix (σ) \cap Const (δ)$ .

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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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

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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

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

Jan Erik Björk (1970)

Annales de l'institut Fourier

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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

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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

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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

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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

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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...