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: Should a political party form a coalition? Voters and math decide

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

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

Devices or divisive: Mobile technology in the classroom

16 hours ago

Little is known about how new mobile technologies affect students' development of non-cognitive skills such as empathy, self-control, problem solving, and teamwork. Two Boston College researchers say it's ...

Forming school networks to educate 'the new mainstream'

22 hours ago

As immigration increases the number of non-English speaking "culturally and linguistically diverse" students, schools will need to band together in networks focused on the challenges of educating what has been called "the ...

Rare tidal movements expose Kimberley dinosaur tracks

22 hours ago

While audiences in Perth attend Walking with Dinosaurs this weekend palaeontologists working near Broome will be documenting the extinct vertebrates' extensive fossilised footsteps using laser scanning technology.

User comments : 1

Adjust slider to filter visible comments by rank

Display comments: newest first

Stephen_Crowley
1 / 5 (3) Jun 10, 2013

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.