Professor Grigori Rozenblioum on Elin Oxenhielm’s Paper

I just received this via email from Grigori Rozenblioum, professor of mathematics, a copy of a letter he has sent to the journal Nonlinear Analysis:

Dear Editors of Nonlinear Analysis,

Probably, this was not meant to be so, but the paper by Elin Oxenhielm, on the 16-th Hilbert problem, to be published soon in ‘Nonlinear Analysis’ has caused an ‘unhealthy’ publicity here in Sweden, in various media. I have downloaded the paper and I have a very strong feeling that it was a heavy mistake to publish it in such a respectable journal.

What is the method of the proof? The author, considering a nonlinear system, supposes that the solutions are ‘dominated by a harmonic term of a specific order’ without defining the word ‘dominated’, and, from what follows later, meaning simply that this is a good approximation, whithout specifying how good approximation is.

Then the author substitutes this approximation into the equation, neglects the terms she does not like and, by considering the remaining terms, declares her statement proved. This lies far below generally accepted methematical standards of proof, since no estimates of the influence of the deleted terms on the behavior of the trajectories are made. Instead of this, the author writes, without any justifications, that the ‘method’ enables one to study the structure of limit circles in most general situation.

I give my humble advice to the Editors of ‘Nonlinear Analysis’ to re-consider the decision on publication of the article by Elin Oxenhielm.

Grigori Rozenblum, Professor of Mathematics,

Chalmers University of Technology,

Göteborg, Sweden