Talk:Hilbert series and Hilbert polynomial

message: This article is really a stub. I have rewritten the introduction, but the new introduction implies to rewrite the remainder of the page. I'll do that, but my other occupations implies that it would need some time. D.Lazard (talk) 18:58, 3 December 2011 (UTC)

message: A new section has recently been added. A discussion is started in User talk:D.Lazard#On Hilbert polynomial, abut its place in the article and its title. It is better to continue it here. D.Lazard (talk) 11:04, 2 January 2017 (UTC)
replies: text: I'm more or less OK with the current title and the position, though I still disagree with the explanation for the change. The main point of a new section is the definition using Euler characteristic of twists of a "geometric object", and it's not so important whether it's stated for coherent sheaves on proper schemes or, say, holomorphic vector bundles on complex projective manifolds. This type of definition is used in complex geometry, in classical algebraic geometry for algebraic varieties, as well as for proper schemes. Thus I don't think it's fair to say it's restricted to a specific area, and certainly not to scheme theory. One of the reasons it's stated for schemes is that Proj construction provides a dictionary which translates the classical definition for graded modules into this definition for coherent sheaves on projective schemes, so it's not even a generalization in this case, but essentially a restatement with fancier words. Anyway, I think the title "Generalization to coherent sheaves" is OK, since, for example, holomorphic vector bundles are coherent analytic sheaves, and the place in the article isn't very important for me, so we may just stop arguing.  — Preceding unsigned comment added by Dpirozhkov (talk • contribs) 16:12, 2 January 2017 (UTC)
thread_title:  Geometric definition/Generalization to coherent sheaves 

message: I think graded free resolutions should be discussed on this page since they can be used to find the hilbert polynomial of a coherent sheaf. There is a nice tutorial article http://www.webpages.uidaho.edu/~abo/Research/smi/sample-solutions5.pdf going over this. Also, there should be code on this page showing how to do these computations with Macaulay2. — Preceding unsigned comment added by 97.122.75.155 (talk • contribs) 05:21, 26 July 2017 (UTC)
thread_title:  Graded Free Resolutions and Hilbert Polynomials 

message: This page should discuss the computation of intersection multiplicity from the hilbert polynomial: https://web.archive.org/web/20170829192231/http://www.math.colostate.edu/~achter/672f06/help/hilb_poly.pdf  https://web.archive.org/web/20170829193032/http://www-personal.umich.edu/~eclader/HilbertPolynomials.pdf — Preceding unsigned comment added by 128.138.65.205 (talk) 19:28, 29 August 2017 (UTC)
thread_title:  Intersection Multiplicity 

message: * Add union of complete intersections
* Expand page to include different weights
This page should include the examples from this article: http://homepages.warwick.ac.uk/~masda/surf/more/grad.pdf Username6330 (talk) 02:32, 26 September 2017 (UTC)
replies: text: This article (by Reid) is specifically about weighted graduations, while these graduations are only sketched in the Wikipedia article. I have not found in it any example that could illustrate the present content of the article. Reid's article could be used for adding more content relative to general graded rings. However, this article is not published, and therefore can hardly be used as a source for Wikipedia. D.Lazard (talk) 09:18, 26 September 2017 (UTC)
replies: text:  The point is to guide wikipedians towards improvement of this encyclopedia. Whether I write this up or someone else does is up to them. There is a significant deficit of useful information on wikipedia. This page could be modified to include this content.Username6330 (talk) 05:58, 29 September 2017 (UTC)
thread_title:  Additional Examples 

message: Does anyone know if there is a nice way to write down combinations as polynomials without having to work them out every time? The motivation for this question is the computations using free resolutions.  — Preceding unsigned comment added by 50.246.213.170 (talk) 00:01, 20 November 2017 (UTC)
thread_title:  Writing down hilbert polynomials from combinations 

message: This section says that (1-t)^d\,HS_R(t) is a polynomial, which is equal to the numerator P(t) of the Hilbert series of . However, the Krull dimension of  is equal to d + 1, therefore HS_R(t)=\frac{P(t)}{(1-t)^{d+1}}. I have also made this edit. Danneks (talk) 13:18, 5 January 2018 (UTC)
replies: text: Good point, at this step, the denominator is still . I'll try fixing this (the result is definitively correct). D.Lazard (talk) 14:14, 6 January 2018 (UTC)

text: I have fixed this point, and another error (the dehomogenization may not always be done with respect to x_0). I have also added some details and comments for making understanding easier. I hope that, now, this is correct and understandable. D.Lazard (talk) 21:37, 8 January 2018 (UTC)
thread_title:  Degree of a projective variety and Bézout's theorem