Hilbert’s 16th Problem

Regarding 22-year old Swedish student Elin Oxenhielm’s alleged solution of the second part of Hilbert’s 16th problem, which is currently making the rounds in the blogosphere—see for instance this BBC News article—I talked to a mathematician friend of mine who, although this isn’t his field of mathematics, says he’s a little skeptic of its validity. He says that she only deals with a simplified version of a part of Hilbert’s problem, and that the proof isn’t rigorous, only formal, and that there are many obvious errors, although minor, and contradictions in her arguments.

But he wants to make a few things clear: First, a simplification isn’t necessarily bad, and it might be possible to generalize it to solve the full problem. Also, formal arguments, he says, can make a good foundation for proving something rigorously (however, there is one article in the list of reference which he currently didn’t have access to, and which might motivate her arguments better). Lastly, minor errors might be nothing more than minor errors, and might not invalidate her argumentation.

I also found this comment on Slashdot, from which I quote:

It’s also categorically not an important theoretical contribution to Hilbert’s 16th problem. […] Incidentally, if this were an important theoretical paper on Hilbert’s 16th problem, the journal “Nonlinear analysis” would be a strange place for it (it’s more interdisciplinary, and is not a mainstream outlet for theoretical mathematics). There’s no reason it couldn’t be true, but it’s some cause for initial suspicion as well as explanation for why the article was accepted. Probably the editors and referees were applied scientists unfamiliar with the problem, who were perfectly happy to accept an approximation justified by some numerical data.

It will be interesting to see how this develops further.