Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square − 1,1 2 . And, we prove that the growth order of the Lebesgue constant is O n + 2 2 . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O n . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square − 1,1 2 , the growth order of which is O ln n 2 .

The geometry of Hermitian self-orthogonal codes

We prove that if $$n >k^2$$ n > k 2 then a k-dimensional linear code of length n over $${\mathbb F}_{q^2}$$ F q 2 has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than n common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.

Asymptotic vectors of entire curves

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t<\infty $, $\left|z\left(t\right)\right|<\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|<\infty \right\}$. We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.

Structure of the set of Borel exceptional vectors for entire curves. II

We have obtained a description of structure of the sets of Picard and Borel exceptional vectors for transcendental entire curve in some sense. We consider only $p$-dimensional entire curves with linearly independent components without common zeros.In particular, the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces in $\mathbb{C}^{p}$ of dimension at most $p-1$. Moreover, the sum of their dimensions does not exceed $p$ if anypairwise intersection of the subspaces contains only the zero vector. A similar result is also valid for the set of Picardexceptional vectors.Another result shows that the structure of the set of Borel exceptional vectors for an entire curve of integer orderdiffers somewhat from the structure of such a set for an entire curve of non-integer order.For a transcendental entire curve $\vec{G}:\mathbb{C}\to \mathbb{C}^{p}$ with linearly independent components and without common zeros having non-integer or zero order the set of Borel exceptional vectors together with the zero vector is a subspace in $\mathbb{C}^{p}$ of dimension at most $p-1$. However, the set of Picard exceptional vectors does not possess this property. We propose two examples of entire curves.The first example shows the set of Borel exceptional vectors together with the zero vector for $p$-dimensional entire curve of integer order isunion of subspaces of dimension at most $p-1$ such that the total sum of these dimensions does not exceed $p$ and intersection of any pair of these subspaces contains only zero vector. The set of Picard exceptional vectors for the curve has the same property.In the second example, we construct a $q$-dimensional entire curve of non-integer order for which the set of Borel exceptional vectors together with the zero vector is a subspace in $\mathbb{C}^{q}$ of dimension at most $q-1$ and the set of Picard exceptional vectors together with the zero vectordo not have the property. This set is a union of some subspaces.

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural “asymptotic gcd” inequality of Pasten and the second author, and consider moving targets versions of our results.