Researcher finds solution to problem in 19th-century theory of meromorphic functions

May 28, 2013
The statement of Gol’dberg conjecture from the original paper.

General theory of meromorphic functions in the complex plane began in the nineteenth century, when E. Picard proved his famous 'Picard's little theorem'. Then, in the 1920s, R. Nevanlinna created the modern theory of meromorphic functions, where his 'second main theorem (SMT)' provides a far-reaching generalization of Picard's theorem. Nowadays, the theory is well-established as a result of many excellent research studies. Nevertheless, the theory still has several unresolved problems, including the following one:

Conjecture of Gol'dberg, middle 1980-s: For every transcendental meromorphic function in the plane, the frequency of distinct poles is governed by the frequency of zeros of the second derivative.

Now, Tokyo Tech Katsutoshi Yamanoi has solved this conjecture.

The solution is based on two important developments in Nevanlinna , which are interesting for their own sake. The first one is a generalization of SMT for small moving targets. The other is a reversion of SMT.

The proof shows that Gol'dberg's conjecture is true in more general form. The results described by Yamanoi in this paper are an important breakthrough in the theory of meromorphic functions.

Explore further: 'Moral victories' might spare you from losing again

More information: Yamanoi, K. Zeros of higher derivatives of meromorphic functions in the complex plane, Proc. London Math. Soc. (2012), 78 pages. DOI:10.1112/plms/pds051

add to favorites email to friend print save as pdf

Related Stories

Mathematician announces that he's proved the ABC conjecture

Sep 12, 2012

(Phys.org)—In all of history there are very few names that stand out in the field of mathematics, at least among those not in the field: Euclid, Newton, Pythagoras, etc. This is likely due to several reasons, chief among ...

Google pays tribute to 'Fermat's Last Theorem'

Aug 17, 2011

Google paid tribute on Wednesday to 17th century French mathematician Pierre de Fermat, transforming its celebrated homepage logo into a blackboard featuring "Fermat's Last Theorem."

Recommended for you

Wyoming cave with fossil secrets to be excavated

5 hours ago

(AP)—For the first time in more than 30 years, paleontologists are preparing excavate a sinkhole-type cave in northern Wyoming that contains the ancient remains of tens of thousands of animals.

Awarded a Pell Grant? Better double-check

16 hours ago

(AP)—Potentially tens of thousands of students awarded a Pell Grant or other need-based federal aid for the coming school year could find it taken away because of a mistake in filling out the form.

User comments : 1

Adjust slider to filter visible comments by rank

Display comments: newest first

Stephen_Crowley
1 / 5 (3) Jun 10, 2013