Talk:Real algebraic geometry

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The section on real plane curves does belong in the article, at present. Please don't just cut it out. Charles Matthews (talk) 10:06, 26 July 2010 (UTC)

I copied the contents of the section Real plane curves to a separate article with the title Real plane curves. I think this is fair. —Preceding unsigned comment added by Jcimpric (talk • contribs) 12:51, 26 July 2010 (UTC)


 * OK, but it needed some attention: real plane curve is the correct title, not the plural form. Charles Matthews (talk) 17:08, 26 July 2010 (UTC)

Edits by user:Estater need to be reverted
Since January 5, user:Estater has edited this page to insert papers by Akbulut in the section Timeline of real algebra and real algebraic geometry. I think that these edits have to be reverted to version of Juny 8, 2012, for the following reasons. D.Lazard (talk) 13:23, 8 January 2012 (UTC) — Preceding unsigned comment added by Remarksen (talk • contribs) 21:42, 8 January 2012 (UTC)
 * WP:advertising, WP:sock puppetry and WP:COI: All the contributions of this user, as well as of those of user:Remarksen an user:Faircool are uniquely devoted to promote Selman Akbulut and his work.
 * WP:neutral: This section is devoted to list the main results which are important for the history of the subject. Having 2/3 of the items since 1980 devoted to Akbulut and his coauthor is clearly a strong historical bias.
 * WP:notability: It seems that none of the inserted items are cited in textbooks on the subject. Most of them seems variants or generalizations of Tognoli's result, thus they do not deserve to be quoted as important dates. The last result inserted is trivial: Every bi-cyclic quartic curve of genus 1 is a real projective non singular curve, and a complex projective singular curve.
 * D.Lazard should consult with the experts first, before making incorrect claims and inflame wiki contributors. For example, the last cited result of Akbulut-King is not about complexification of a given real algebraic set, it is about existence of a smooth submanifold of RP^n which can never be isotoped to the real part of a nonsingular complex algebraic subset of CP^n, even though itself can be isotoped to a nonsingular algebraic subset of RP^n (I modified Estaters edit to emphasize that point). It is well known to the experts that Akbulut-King have been major contributors to the field of Topology of Real Algebraic sets.


 * Before Remarksen edit, the statement about 2005 paper was trivial. After that edit, it is no more trivial. Its notability remains questionable, as non sourced assertions like "It is well known to the experts" are not acceptable by Wikipedia (see WP:OR). - D.Lazard (talk) 22:20, 8 January 2012 (UTC)


 * If you consult to experts, you will find out that there are two major books on "Topology of Real Algebraic Sets" (both listed in the references section), you can consult them. Also missing from "timeline of real algebraic geometry" section is the progress made to Hilbert's 16'th problem (e.g. work of Arnold and Rochlin school), such as "which curve configurations in RP^2 can be realized as algebraic sets of certain degree?". Hopefully another editor will fill this void (you can view the last cited result of Akbulut-King as a certain generalization of this, where the curve configurations in RP^2 is replaced by a smooth manifold in RP^n and the degree restriction is lifted). — Preceding unsigned comment added by Remarksen (talk • contribs) 23:28, 8 January 2012 (UTC)


 * Actually the article in general relies much too heavily on primary sources and the timeline section in particular seems to be a indiscriminate listing of research results. I'd say try to find secondary sources for each bullet in the timeline and remove the ones where they can't be found. There are clear COI issues here but it's not entirely clear this is self promotion or a good faith effort to complete coverage on the subject. In any case, the "If you're not an expert on the subject then you can't undo my edits" argument carries no weight with me at all. If the subject is so abstruse that only a few experts can evaluate the sources then that causes WP:RELIABILITY issues and perhaps an AfD is in order.--RDBury (talk) 13:39, 9 January 2012 (UTC)
 * MathSciNet figures are not completely reliable but are an indication of an article's influence. Thus, the article by Collins, George E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Automata theory and formal languages (Second GI Conf., Kaiserslautern, 1975), pp. 134–183. Lecture Notes in Comput. Sci., Vol. 33, Springer, Berlin, 1975 is cited over 80 times in mathscinet.  I checked a few of Akbulut's papers found in the timeline and found that, while some of them are clearly influential, others fall far short of a standard of notability set by Collins' paper.  Perhaps one can use mathscinet figures as a rule of thumb for including items in the timeline if no other method can be found.  Tkuvho (talk) 13:54, 9 January 2012 (UTC)
 * Consider, for instance, consecutive footnotes 50 and 51, containing respectively 50: S. Akbulut and H. King, Algebraicity of Immersions, Topology, vol. 31, no. 4, (1992), 701-712 and 51: C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039--1069. Akbulut-King appeared eight years earlier than Scheiderer.  However, Akbulut-King fetches only 3 cites in mathscinet, whereas Scheiderer fetches 38.  There is clearly an imbalance.  I would suggest deleting this particular reference by Akbulut-King unless there are other reasons to consider it particularly influential.  Tkuvho (talk) 14:23, 9 January 2012 (UTC)


 * I would say in that field they proved so many ultimate results that it left little room for improvement, hence less people working to improve them and less citations (a similar example is W. P. Thurston papers, last 40 years they collected hundredths of citations, while his two ultimate papers on foliations, which basically finished the whole field, got very few). Also people usually cite Akbulut-King book for those results, rather than the original technical journal versions. Finally, I find you comparing them to citations of papers on sums of squares or automata theory not convincing, they are apples and oranges.  — Preceding unsigned comment added by Remarksen (talk • contribs) 16:04, 9 January 2012 (UTC)
 * The paper by Collins is not on automata theory. It happens to have appeared in a journal by that name, but the theme of the paper appears to be an exploration of the deep connection between logic and real algebraic geometry.  Tkuvho (talk) 16:47, 9 January 2012 (UTC)
 * About the number of citations in MathScinet: Akbulut-King book of 1992 has 48 citations. Bochnak-Coste-Roy book of 1998 has 328 citations. Bochnak-Coste-Roy paper of 1996 entitled "On the combinatorial and algebraic complexity of quantifier elimination" has 48 citations and is not cited in the list. Tognoli's review in MathSciNet on Akbulut-King book says (Tognoli, which appears in the list, is probably the best expert on the subject of studies of Akbulut and King): This book is almost entirely dedicated to the following problem: Give a topological characterisation of the real algebraic sets and The results contained in these chapters [III to VI] are entirely due to the authors (Chapter I is the introduction, II is about generalities and VII, the last on is on dimension 3). The number of citations, as well as Tognoli's review shaw that this book is more a monography than a text book. My proposition: Regroup all Akbulut-King items into a single item referring to their book. It could be Topological characterisation of the real algebraic sets, but a better one line summary may probably be found. This would respect the necessary equilibrium between the various aspects of real algebraic geometry. D.Lazard (talk) 17:41, 9 January 2012 (UTC)
 * That seems to be a reasonable suggestion, especially given the underwhelming citation statistics for a textbook that Remarksen alleges "people usually cite".  Sławomir Biały  (talk) 12:36, 10 January 2012 (UTC)
 * Seems very reasonable. We could perhaps also retain those articles that have a mathscinet count of at least 20 references, say.  Tkuvho (talk) 12:58, 10 January 2012 (UTC)
 * If you use that logic you have to wipe of the two most important theorems of Foliation theory by W.P. Thurston from the Foliations page of wiki, because they got only 14, 18 citations (these ultimate theorems practically finished the field). I find it unreasonable editors trying to enforce balance between practically unrelated subjects from logic, model theory, quadratic forms, ordered fields to the "topology of real algebraic sets". If the prominence of this field in certain period bothers editors, then maybe they should start a new wiki page under "Topology of real algebraic sets" and move Nash, Tognoli, and Akbulut-King contributions to that page (perhaps along with the developments relating to Hilbert's 16th problem). BTW there is also 1936 contribution of Seifert predating Nash, which should have ben mentioned.  — Preceding unsigned comment added by Remarksen (talk • contribs) 15:05, 10 January 2012 (UTC)


 * Today too many different subjects are bundled up under "real algebraic geometry", from logic and model theory, quadratic forms an ordered fields, to the "topology of real algebraic sets" (namely the subject of which topological spaces can be made real algebraic sets, and which topological invariants can be obtained from real algebraic sets), which is the topic of discussion here. — Preceding unsigned comment added by Remarksen (talk • contribs) 18:14, 9 January 2012 (UTC)
 * No, the topic of discussion is not the "topology of real algebraic sets", but its place inside real algebraic geometry and the place of Akbulut-King inside "topology of real algebraic sets". A good source to judge of these questions is certainly Bochnak-Coste-Roy book whose review MR0949442 in MathSciNet says "the book under review is the first text trying to bring together all basic problems and techniques in the field" and "the book is one of the finest ever read by this reviewer, who strongly recommends it to anyone wishing to appreciate the intricate beauty of real algebraic geometry". The review MR1659509 of the English edition says The most useful new information is contained in the "Bibliographic and historical notes" at the end of each chapter. Among 15 chapters, Algebraic models of smooth manifolds is the 14th, and Akbulut-King work is treated in its second section, which exactly supports my proposition.
 * By the way, this book and Basu-Pollack-Roy one, which completes it for the algorithmic aspects, are a unique secondary source for the whole area of real algebraic geometry. I suggest thus to use them as the main source to improve the "guideline section" and, more generally, the whole page.
 * D.Lazard (talk) 16:39, 10 January 2012 (UTC)
 * You can not judge the importance of a subject in which chapter of some book it is mentioned (BTW thanks for the good PR job for Bochnak-Coste-Roy book). That book choses to emphasize algebra more than topology, then at the end mentions topological contributions, whereas Akbulut-King emphasis is purely on "topology of real algebraic sets",  it has much deeper far reaching theorems in that subject.  Rather than bickering about which aspects are more important (pure algebra verses topology and geometry) I suggest you start a  new wiki page  under "Topology of real algebraic sets".    — Preceding unsigned comment added by Remarksen (talk • contribs) 17:30, 10 January 2012 (UTC)
 * A page Topology of real algebraic sets would certainly be welcome in Wikipedia and you are certainly the best Wikipedia editor for writing it. When it will exist, it will naturally be linked to from Real algebraic geometry and certainly from some pages of Category:Topology. — D.Lazard (talk) 18:11, 10 January 2012 (UTC)
 * That seems like a reasonable solution. The page can be created simply by clicking on the red link above.  Then you can start typing!  Tkuvho (talk) 18:14, 10 January 2012 (UTC)
 * Sure I can do that, e.g. start with moving Hilbert, Harnack, Petrovski-Olenik, Seifert, Nash, Tarski-Seidenberg, Hironaka, Lojasiewicz, Milnor, Thom, Sullivan, Tognoli, Akbulut-King contributions there + adding the work of the Russian school (e.g. Rohlin, Arnold, Viro, Kharlamov, Mikhalkin, Itenberg), then wait for further additions. Of course it would be nice to have an enforcer like D.Lazard there to verify references (maybe he can do that job there too). Also it would be nice if somebody creates a "Topology of real algebraic sets" subsection to the "Real algebraic geometry" page, explaining what that field is about with a short summary and linking to the main "Topology of real algebraic sets" page. If there is an agreement I can at least take the initial steps.
 * I now realize that there are already 3 wiki pages relating to Hilbert's 16th problem (mostly about the work of Russian school): "Hilbert's 16'th problem", " Harnack's curve theorem" and "Ragsdale conjecture". This would make creating "Topology of real algebraic sets" page easy (about those subjects just link to those 3 pages).


 * After some thoughts, I now realize the potential risks of being the person who first activates the new "topology of real algebraic sets" page, such as being seen as the one who caused Akbulut-King's ouster from this "real algebraic geometry" page (not to mention collecting criticisms for the omissions on that new page). I already got a flak here for moving into an argument, and quickly accused of being Akbulut promoter, now I don't want to be accused by the opposite side for yanking their stuff off the real algebraic geometry page. I feel when assigning credits to decades long of people's work are concern one should proceed cautiously. But if this "topology of real algebraic sets" page gets used by other editors and strarts developing, then I could consider contributing it.  — Preceding unsigned comment added by Remarksen (talk • contribs) 05:51, 11 January 2012 (UTC)
 * — Preceding unsigned comment added by Remarksen (talk • contribs) 19:23, 10 January 2012 (UTC)

Artin-Schreier
The Artin-Schreier paper of 1926/7 seems to be the foundation stone of real closed fields. Should it be in the timeline? Tkuvho (talk) 17:19, 9 January 2012 (UTC)
 * Probably. This timeline is clearly a stub which contains omissions. By the way, it has been suggested to provide secondary references to the items. I believe that Bochnak-Coste-Roy and Basu-Pollack-Roy books are such a secondary reference for most of them. This has to be checked. D.Lazard (talk) 17:48, 9 January 2012 (UTC)