For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on $L_p\left(\mu \right)$, 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.

We construct a sequence of homogeneous polynomials on the unit ball ${}_d$ in ${ℂ}^d$ which are big at each point of the unit sphere . As an application we construct a holomorphic function on ${}_d$ which is not integrable with any power on the intersection of ${}_d$ with any complex subspace.

We study the problem of whether ${}_w\left(ⁿE\right)$, the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if ${}_w\left(ⁿE\right)$ is an M-ideal in (ⁿE), then ${}_w\left(ⁿE\right)$ coincides with ${}_{w0}\left(ⁿE\right)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if ${}_w\left(ⁿE\right){=}_{w0}\left(ⁿE\right)$ and (E) is an...

Let $\parallel ·\parallel$ be the uniform norm in the unit disk. We study the quantities $M_n\left(\alpha \right):=inf(\parallel zP\left(z\right)+\alpha \parallel -\alpha )$ where the infimum is taken over all polynomials $P$ of degree $n-1$ with $\parallel P\left(z\right)\parallel =1$ and $\alpha >0$. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{\alpha >0}{M}_n\left(\alpha \right)=1/n$. We find the exact values of $M_n\left(\alpha \right)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Let K be any subset of ${ℂ}^N$. We define a pluricomplex Green’s function $V_{K,\theta }$ for θ-incomplete polynomials. We establish properties of $V_{K,\theta }$ 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 $K\setminus supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$. We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$ when K is a compact...

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.

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.

We consider two operator space versions of type and cotype, namely $S_p$-type, $S_q$-cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_p$ spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little...

Letting $T_n$ (resp. $U_n$) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ${\left(X^k{T}_{n-k}\right)}_k$ and ${\left(X^k{U}_{n-k}\right)}_k$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space ${}_n\left[X\right]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_n$ and $U_n$ admit remarkableness integer coordinates on each of the two basis.

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

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r{(1-x)}^r$, r ≥ 0, and let $P_{n,r}^{\mathbb{Z}}\subset P_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^{\mathbb{Z}}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (P_{n,r}^{\mathbb{Z}};L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ for p ≠ 2. It follows that $\mu (P_{n,r}^{\mathbb{Z}};L_p)\asymp n^{-2r-2/p}$ as n → ∞. The results...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Let ${}_q\left[t\right]$ be the polynomial ring over the finite field ${}_q$, and let ${}_N$ be the subset of ${}_q\left[t\right]$ containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set $A{\subseteq }_N$ for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that $D\left(N\right)\ll q^N{\left(logN\right)}^7/N$.

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ${ₙ}^c$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ\in {ₙ}^c$ of the form $Pₙ\left(z\right):={\sum }_{j=0}^n{a}_j{z}_j$, $a_j\in C$, by $Sₙ\left(Pₙ\right)\left(z\right):={\sum }_{j=0}^{n}a{̃}_j{z}_j$, $a{̃}_j:=a_j\left|a_j\right|min|a_j|,1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{real}:=su{p}_{Pₙ\in ₙ}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|)$, $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{comp}:=su{p}_{Pₙ\in {ₙ}^c}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for...

If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k\le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)}(\alpha )\ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it...

If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k\le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)}(\alpha )\ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it...

The extremal functions  $f_0\left(z\right)$  realizing the maxima of some functionals (e.g. $max|a_3|$, and  $maxarg{f}^{{}^{\text{'}}}\left(z\right)$) within the so-called universal linearly invariant family $U_{\alpha }$ (in the sense of Pommerenke [10]) have such a form that $f_0^{{}^{\text{'}}}\left(z\right)$  looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials $P_n^{\lambda }(x;\theta ,\psi )$ of a real variable $x$ as coefficients of $$G^{\lambda }(x;\theta ,\psi ;z)=\frac{1}{{(1-z{e}^{i\theta })}^{\lambda -ix}{(1-z{e}^{i\psi })}^{\lambda +ix}}=\sum _{n=0}^{\infty }{P}_n^{\lambda }(x;\theta ,\psi )z^n,\left|z\right|<1,$$ where the parameters $\lambda$, $\theta$, $\psi$ satisfy the conditions:...

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 $1\le j\le d$ 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...

We apply pluripotential theory to establish results in ${ℝ}^k$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $\Sigma \subset {ℝ}^k$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

It is well known that if φ(t) ≡ t, then the system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is a basis in some Lebesgue space $L_p$. The aim of this short note is to show that the answer to this question is negative.

We show that a Banach space X is an ℒ₁-space (respectively, an ${ℒ}_{\infty }$-space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space ${}_{wb}{(}^{m}X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an ${ℒ}_{\infty }$-space.