Revision as of 13:46, 3 May 2006 edit Lambiam (talk \| contribs) Autopatrolled, Extended confirmed users, Page movers 65,058 edits Gröbner basis and Hironaka ← Previous edit		Revision as of 07:29, 5 May 2006 edit undo LutzL (talk \| contribs) Extended confirmed users 1,187 edits →Gröbner basis and Hironaka Next edit →
Line 27: Hi. Concerning your recent edit of [[Gröbner basis]], I am not sure that Hironaka's theory of "standard bases" is exactly the same. See for example [http://lips.informatik.uni-leipzig.de/pub/showDoc.Fulltext/dokument.pdf?lang=de&doc=1995-18&format=pdf&compression=&rank=0 Joachim Apel, Division of entire functions by polynomial ideals, in ''Proc. AAECC 11,'' LNCS 948 (1995), pp. 82-95]. So if you agree but think the addition is nevertheless important, I propose that you change the text into something like: "In 1964, at almost the same time and independently, [[Heisuke Hironaka]] had developed a closely related theory, which he called '''standard bases'''." I'm not sure how close this is to your areas of expertise, but I you have interest, time and patience, perhaps you could also add some of this to the Hironaka article, putting it in context, and if possible cite the references as provided by Apel's paper. Cheerio. [[User:Lambiam\|Lambiam]][[User talk:Lambiam\|<small><sup>Talk</sup></small>]] 13:46, 3 May 2006 (UTC) :As I understand it now, the standard bases are defined for ideals in the ring of Puisseux series. Thus they contain Gröbner bases as a special case. The real contribution of Buchberger to the fundamentals is the proof that his algorithm stops in finite time. Perhaps it should be mentioned somewhere that the idea comes from the non-constructive proof by Hilbert of the Basissatz (Kronecker had a constructive proof, but only formulated for bivariate polynomials) -- No, I'm not an expert in the whole of the topic of Gröbner bases. They are interesting to my in the aspect of their inefficiency. I got most of my knowledge from ''Gethgen/vonzur Gathen: Modern Algebra'', where they mention H. Hironaka in the second sentence of the introduction. And from ''M. Guisty: Bases standard, élimination et complexité'', notes of a lecture given at [http://www.math.polytechnique.fr/xups/programme97.html X], where some propositions are attributed to Hironaka.--[[User:LutzL\|LutzL]] 07:29, 5 May 2006 (UTC)

LutzL