Talk:Hilbert's sixteenth problem

Introspective

This rather introspective chunk was posted: As of the curves of order 6, I have - admittedly in a rather elaborate way - convinced myself that the 11 branches, that they have according to Harnack, never all can be seperate, rather there must exist with one branch running in its interior and nine branches running in its exterior, or opposite. It seems to me that a thorough investigation of the relative positions of the upper bound for seperate branches is of great interest, and similarly the corresponding investigation of the number, shape and and position of the sheets of an algebraic surface in space - it is not yet even known, how many sheets a surface of order 4 in three-dimensional space can maximally have. (cf. Rohn, Flächen vierter Ordnung, Preissschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig 1886)

Could someone can decode that into something less personal and more encyclopedic, if possible? Thanks Dysprosia 10:27, 28 Nov 2003 (UTC)

That was a translation of Hilbert's original problem. That is, the I in the translation is Hilbert. A transcription of the problem into more clear prose might be desirable, but should not replace Hilbert's original statement of the problem.

So was this a quotation? It just didn't seem clear enough that it was Dysprosia 10:31, 28 Nov 2003 (UTC)

Added quotes to emphasize that it is a quotation.

Thank you :) Dysprosia 10:38, 28 Nov 2003 (UTC)

Hilbert's original statement of the problem was in German. What is reproduced here is a translation. The most widely used English translation of Hilbert's text is the translation by Mary Winston Newson published in 1902. The

HTML version of this translation provided by David Joyce (mentioned under external links in the main Hilbert's problems article) is widely referenced on the web, and pops up quickly if you do a Google search on "Hilbert problems". I don't think reproducing an alternative (and less clear) English translation of the original Hilbert text in Wikipedia adds any value, plus there is a Wikipedia style guideline that says don't include copies of primary sources. Much more useful, to my mind, would be an article summarising the history and current status of the 16th problem. -- Gandalf61 14:38, Nov 28, 2003 (UTC)

I agree that the page needed more than a pure translation. I have tried to reformulate the page with more modern language and references to the history of the problem(s). Some images would probably help a lot for the understanding of the first part, though.

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The following does not make sense:

Here we are going to consider polynomial vector fields in the plane, that is a system of differential equations of the form:

${dx \over dt}P(x,y),\qquad {dy \over dt}Q(x,y)$

where both P and Q are polynomial functions of degree n.

If it says "differential equations of the form ..." then what follows needs to be a differential equation. What is displayed is not a differential equation. I'd fix it if I knew what it should say instead. Michael Hardy 21:04, 2 Mar 2004 (UTC)

• Maybe, it should be "differential expressions" instead of "differential equations"? Andris 21:07, 2 Mar 2004 (UTC)

Potential solution to the second part of the problem

There is a fairly recent article proposing a solution to the second half of this problem. Should it be mentioned? 2804:7F0:7780:9B0:18ED:B799:B2F:6707 (talk) 19:00, 30 October 2024 (UTC)

The pre-print "A note on a recent attempt to solve the second part of Hilbert's 16th Problem" presented some counterexamples to this article. Saung Tadashi (talk) 23:53, 18 November 2024 (UTC)