STRUCTURES OF THE SETS OF BOREL AND PICARD EXCEPTIONAL VECTORS OF ANALYTIC CURVES

Authors

  • Yaroslav Savchuk Ivano-Frankivsk National Technical University of Oil and Gas

DOI:

https://doi.org/10.31471/2304-7399-2023-18(68)-18-24

Keywords:

entire curve, analytic curve in the unit disc, the Picard exceptional vector, the Borel exceptional vector.

Abstract

Previously, we have described the structure of the sets of Picard and Borel exceptional vectors for a transcendental p-dimensional integer curve with linearly independent components without common zeros. It was proved that the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces of dimension not higher than p-1 of a p-dimensional complex Euclidean space. In addition, the sum of the dimensions of all these subspaces does not exceed p and any pairwise intersection of these subspaces contains only the zero vector. The set of Picard exceptional vectors has the same structure. For an integer curve of non-integer or zero order, the set of Borel exceptional vectors together with the zero vector is a single subspace of dimension no higher than p-1, which is not the case for the set of Picard exceptional vectors of an integer curve of non-integer order. The sufficiency of these results is confirmed to some extent by the corresponding examples.  In this paper, we show that the above results on the structure of the set of Borel and Picard exceptional vectors for integer curves can be transferred to the case of analytic curves in the circle. We have also constructed an analytic curve in the circle, which confirms, to some extent, the sufficiency of the results. However, it was not possible to find out whether the corresponding result is transferred to the case of analytic curves of non-integer or zero order.

References

Weyl H., Weyl J. Meromorphic functions and analytic curves. – Prinseton: Prinseton Univ. Press, 1943. – 531 p.

Петренко В.П. Целые кривые. Х.: Вища школа, 1984. – 136 с.

Savchuk Ya. I., Structure of the set of defect vectors of entire and analytic curves of finite order, Ukr. Math. J., 37 (1985), №5, 494–499.

Savchuk Ya.I., Inverse problem of the theory of distribution of the values of entire and analytic curves, Journal of Soviet Mathematics, 48 (1990), №2, 220–231.

A.I. Bandura, Ya.I. Savchuk, Structure of the set of Borel exceptional vectors for entire curves, Matematychni Studii 53 (1), 41-47. doi: 10.30970/ms.53.1.41-47

Ya.I. Savchuk, A.I. Bandura, Structure of the set of Borel exceptional vectors for entire curves. II Mat. Stud. (2021), Vol. 55 (2), 137-145. doi: 10.30970/ms.55.2.137-145.

Published

2023-12-01

How to Cite

Savchuk, Y. (2023). STRUCTURES OF THE SETS OF BOREL AND PICARD EXCEPTIONAL VECTORS OF ANALYTIC CURVES. PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY Number, (18(68), 18–24. https://doi.org/10.31471/2304-7399-2023-18(68)-18-24