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.

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 X be an analytic set defined by polynomials whose coefficients $a₁,...,a_s$ are holomorphic functions. We formulate conditions on sequences $a_{1,\nu },...,a_{s,\nu }$ of holomorphic functions converging locally uniformly to $a₁,...,a_s$, respectively, such that the sequence $X_{\nu }$ of sets obtained by replacing $a_j$’s by $a_{j,\nu }$’s in the polynomials converges to X.

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.

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...

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...

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 $D\left(N\right)\ll q^N{\left(logN\right)}^7/N$.

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...

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.

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

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:...

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 $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

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...

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...

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...

Let $𝔻$ denote the unit disk $\{z:|z|<1\}$ in the complex plane $ℂ$. In this paper, we study a family of polynomials $P$ with only one zero lying outside $\overline{𝔻}$.  We establish  criteria for $P$ to satisfy implying that each of $P$ and $P^{\text{'}}$  has exactly one critical point outside $\overline{𝔻}$.

Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by $T^{†}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $\Phi \left(A\right)\Phi \left(B\right)=\Phi \left(B\right)\Phi {\left(A\right)}^{†}$ for all A, B ∈ ℬ(H) with $AB=B{A}^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $\Phi \left(A\right)=cUA{U}^{†}$ for all A ∈ ℬ(H).