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

  • A.I. Bandura Ivano-Frankivsk National Tecnical University of OIl and Gas
  • Ya.I. Savchuk Ivano-Frankivsk National Technical University of Oil and Gas
Keywords: entire curve, integral curve, Borel exceptional vector, Nevanlinna deficient vector, meromorphic function, admissible system of vectors, Picard exceptional vector, Picard exceptional value

Abstract

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 any
pairwise intersection of the subspaces contains only the zero vector.
A similar result is also valid for the set of Picard
exceptional vectors.
Another result shows that the structure of the set of Borel exceptional vectors for an entire curve of integer order
differs 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 is
union 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 vector
do not have the property. This set is a union of some subspaces.

References

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Published
2021-06-22
How to Cite
1.
Bandura A, Savchuk Y. Structure of the set of Borel exceptional vectors for entire curves. II. Mat. Stud. [Internet]. 2021Jun.22 [cited 2021Oct.19];55(2):137-45. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/190
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