BOREL EXCEPTIONAL VECTORS FOR ENTIRE CURVES WITH LINEARLY DEPENDENT COMPONENTS
DOI:
https://doi.org/10.31471/2304-7399-2024-19(73)-24-32Keywords:
entire curve with linearly dependent components, Nevanlinna’s exceptional vector, Valiron exceptional vector, admissible system of vectors, meromorphic function.Abstract
The paper is devoted to the description of the structure of the set of the Borel exceptional vectors for a transcendental entire curve with linearly dependent components without common zeros. The structure of the set of the Picard and the Borel exceptional vectors for a transcendental p-dimensional integer curve with linearly independent components without common zeros was previously described. In particular, it was established that in a p-dimensional complex space the number of the Borel exceptional vectors admissible in this space can be no more than p. It was proved that the set of the Borel exceptional vectors together with the zero vector is a finite union of subspaces of dimension no higher than p-1 of the 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. It is also known that the structure of the set of Nevanlinna exceptional vectors for a finite-order entire curve with linearly dependent components is similar to that of an ordinary finite-order entire curve.
In the proposed paper, we show that for an integer curve with linearly dependent components, the number of admissible Borel exceptional vectors can be greater than the dimension of the space, but not greater than a certain number, which depends on the dimension of the space and the degree of dependence of the components.
It is also proved that the set of the Borel exceptional vectors in combination with the vectors orthogonal to the entire curve and the zero vector can be represented as a finite union of subspaces of dimension no higher than p-1.
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