On Oka's Extra-Zero Problem
Makoto Abe, Sachiko Hamano, Junjiro Noguchi
TL;DR
This paper provides a comprehensive solution to Oka's extra-zero problem, which asks whether arbitrary Cousin II problems can be solved by adding disjoint zeros, including examples and new questions.
Contribution
It offers a complete resolution of Oka's extra-zero problem, extending previous partial results and counterexamples, and discusses related open questions.
Findings
Complete solution to Oka's extra-zero problem
Counterexamples in higher dimensions
New related questions proposed
Abstract
After the solution of Cousin II problem by K. Oka III in 1939, he thought an {\it extra-zero problem} in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and to discuss some new questions.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Meromorphic and Entire Functions · Mathematics and Applications