Nonreciprocal algebraic numbers of small Mahler's measure

Artūras Dubickas; Jonas Jankauskas

Displaying similar documents to “Nonreciprocal algebraic numbers of small Mahler's measure”

Factorization of irreducible polynomials over a finite field with the substitution $x^{(} q^{r}) - x$ for x

Andrew Long (1973)

Acta Arithmetica

Similarity:

A class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

Similarity:

Let $f (x) = x ⁿ + k_{n - 1} x^{n - 1} + k_{n - 2} x^{n - 2} + \dots + k ₁ x + k ₀ \in ℤ [x]$ , where $3 \leq k_{n - 1} \leq k_{n - 2} \leq \dots \leq k ₁ \leq k ₀ \leq 2 k_{n - 1} - 3$ . We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of $2 k_{n - 1} - 3$ on the coefficients of f(x) is the best possible in this situation.

Polynomial relations amongst algebraic units of low measure

John Garza (2014)

Acta Arithmetica

Similarity:

For an algebraic number field and a subset $α_{1}, . . ., α_{r} \subseteq_{}$ , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $ℚ [x_{1}, . . ., x_{r}]$ vanishing at the point $(α_{1}, . . ., α_{r})$ .

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae

Similarity:

Let $P_{k}$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_{k}$ in $f = \sum_{k} a_{k} P_{k}$ . A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_{k}$ results in obtaining a low order of the recurrence.

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let $∥ \cdot ∥$ be the uniform norm in the unit disk. We study the quantities $M_{n} (α) : = inf (∥ z P (z) + α ∥ - α)$ 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 ${inf}_{α > 0} M_{n} (α) = 1 / n$ . We find the exact values of $M_{n} (α)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

The factorization of $f (x) x^{n} + g (x)$ with $f (x)$ monic and of degree $\leq 2$ .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

In this paper we investigate the factorization of the polynomials $f (x) x^{n} + g (x) \in ℤ [x]$ in the special case where $f (x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f (x)$ is monic and linear.

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

Similarity:

Let K be any subset of $ℂ^{N}$ . We define a pluricomplex Green’s function $V_{K, θ}$ for θ-incomplete polynomials. We establish properties of $V_{K, θ}$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of $ℝ^{N}$ , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K ∖ s u p p {(d d^{c} V_{K, θ})}^{N}$ . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $s u p p {(d d^{c} V_{K, θ})}^{N}$ when K is a compact...

Inequalities for two sine polynomials

Horst Alzer, Stamatis Koumandos (2006)

Colloquium Mathematicae

Similarity:

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $α \leq \sum_{j = 1}^{n - 1} 1 / (n ² - j ²) s i n (j x) \leq β$ , with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0 < \sum_{j = 1}^{n - 1} (n ² - j ²) s i n (j x)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján

Similarity:

The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD $(f, g)$ ) can be obtained from the Sylvester subresultant matrix $S_{j} (f, g)$ transformed to lower triangular form, where $1 \leq j \leq d$ and $d =$ deg(GCD $(f, g)$ ) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_{j} (f, g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD $(f, g)$ is formulated. If inexact polynomials are given, then an approximate...

Characterization of functions whose forward differences are exponential polynomials

J. M. Almira (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Given ${h_{1}, \dots, h_{t}}$ a finite subset of $ℝ^{d}$ , we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $ℝ^{d}$ with the property that the forward differences $Δ_{h_{k}}^{m_{k}} f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_{1}, \dots, m_{t}$ .

A generalisation of Amitsur's A-polynomials

Adam Owen, Susanne Pumplün (2021)

Communications in Mathematics

Similarity:

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

Lower bounds for norms of products of polynomials on $L_{p}$ spaces

Daniel Carando, Damián Pinasco, Jorge Tomás Rodríguez (2013)

Studia Mathematica

Similarity:

For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on $L_{p} (μ)$ , whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in infinite-dimensional settings). The result also holds for the Schatten classes $_{p}$ . For p > 2 we present some estimates on the constants involved.

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

Similarity:

We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

The image of multilinear polynomials evaluated on $3 \times 3$ upper triangular matrices

Thiago Castilho de Mello (2021)

Communications in Mathematics

Similarity:

We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3 \times 3$ upper triangular matrix algebra over an infinite field.