Talk:Hilbert series and Hilbert polynomial

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)

Geometric definition/Generalization to coherent sheaves
A new section has recently been added. A discussion is started in User talk:D.Lazard, 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)
 * 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)

Graded Free Resolutions and Hilbert Polynomials
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)

Intersection Multiplicity
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)

Deletion of Graded Free Resolutions
The section on graded-free resolutions should not have been deleted! They are very important tools for computing hilbert polynomials since they can be readily determined in general (even by hand for nice cases). The computation giving this example should be returned. Check out https://web.archive.org/web/20170922224447/http://www.math.ualberta.ca/~xichen/math68216w/hilbert_polynomial.pdf for more information (such as the hilbert syzygy theorem and the resulting computations). — Preceding unsigned comment added by 73.181.114.81 (talk • contribs) 22:45, 22 September 2017 (UTC)
 * Please, sign your posts in talk page with four tildes ( ~ ).
 * The section on free graded resolutions has not be deleted. It has been moved and edited.
 * Your assertion that free resolutions are a tool for computing Hilbert polynomials is wrong: All known algorithms for computing free resolutions, Hilbert polynomials, and Hilbert series start by the computation of a Gröbner basis, which is the same for the three problems. From this Gröbner basis, the Hilbert series is easier to compute directly than from a free resolution. Thus free resolutions are never used in algorithms for Hilbert series and Hilbert polynomials.
 * It is wrong that free resolutions can be readily determined in general, as Castelnuovo–Mumford regularity, which measures the degrees appearing in a minimal free resolution, may be doubly exponential in the number of indeterminates. The only cases where a free resolution is easy to compute is when the ideal is generated by a regular sequence, and in this case, there is an explicit formula for the Hilbert series, which is given in the article. Thus, even in this case free resolutions are not useful for computing Hilbert polynomial.
 * However, I agree that the relations between Hilbert series, Hilbert polynomials and free resolutions are theoretically important, and was lacking in this article. This is for this reason that I have used your edit for filling this gap.
 * The references that you provide cannot be used in Wikipedia, as they are not reliably published (see WP:OR). D.Lazard (talk) 16:16, 23 September 2017 (UTC)
 * It may be that computer algebra systems only use grobner bases to compute the hilbert polynomial, but there are many examples which can be computed by hand. Consider the non-complete intersection ideal $$I=(xz,yz) \subset \mathbb{C}[x,y,z,w]=R$$. There is an easy resolution of $$R/I$$ given by

0 \to R(-3) \xrightarrow{\begin{bmatrix} y \\ -x \end{bmatrix}} R(-2)\oplus R(-2) \xrightarrow{(xz, yz)} R/I \to 0 $$
 * You can use the product of complete intersection ideals to come up with many non-trivial computations. Did you look at the links I put up?
 * The point is to let editors have a resource for doing computations; I do agree that the correct sources should be cited, but the information presented here is super useful. — Preceding unsigned comment added by 75.166.193.229 (talk • contribs) 24 September 2017 (UTC)
 * Again, please, sign your posts in talk page with four tildes ( ~ ).
 * Please, do not edit other's posts. Interleaving comments in other posts may change the meaning and makes difficult for others to recognize the author of each sentence, and to understand the chronology of the postings. For this reason, I have moved the first part of your post after my post. D.Lazard (talk) 01:17, 25 September 2017 (UTC)

Additional Examples

 * 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)
 * 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)
 * 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)

Writing down hilbert polynomials from combinations
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)

New section "Computing Hilbert Polynomials with Free Resolutions"
An IP editor has recently created this section again, after having reverted several times. The edit summary is "There is no dispute, we already discussed how free resolutions can be found by hand for lot's of non-complete intersection examples. I don't see the issue". This is blatantly wrong, as this new section has been reverted several times, and I have explained why, in edit summaries and in this talk page.

Thus, these edits are against the guideline WP:BRD, since the IP editor has started an edit war, without trying to reach a consensus. On the other hand, for trying to resolve the dispute, I have edited the article by inserting a section "Relation with free resolutions". Apparently, the IP has not read this new section, as its content is completely ignored by his additions.

WP:BRD asserts clearly that, in such a case, the editor, who has boldly edited and has been reverted, must wait for a consensus for editing again. For this reason I'll revert again IP's edits. Nevertheless, I'll detail below the reasons for which they cannot be accepted in WP.

Reasons for not accepting the new section: D.Lazard (talk) 11:39, 1 December 2017 (UTC)
 * Systematic use, without definition nor link, of very technical notations that are not useful here, and may be confusing for most readers ($$\text{Proj}, \chi, \mathcal O, \ldots$$)
 * Use of different notations than elsewhere in the article, typically for the Hilbert polynomial itself
 * Capitalization of the headers that does not follow MOS:HEADCAPS
 * Confusing use of m for the indeterminate of the Hilbert polynomial
 * Wrong main header: these are not about "computing Hilbert polynomials", since computing a Hilbert polynomial is generally easier (and never harder) than computing a free resolution
 * Misplaced: this is nothing more than examples aimed to illustrate the section "Relation with free resolutions". Thus this may appear as a subsection "Examples" of this section, but this requires a complete rewriting for harmonizing terminology and notation.

Rebuttle
Username6330 (talk) 02:17, 2 December 2017 (UTC)
 * 1) Instead of deleting the material why don't you provide links to the relevant pages? They are no more complex than the graded rings/modules themselves. Moreover, it let's the reader know there is a more general definition of the hilbert polynomial using homological algebra/the euler characteristic. The point is that there is a morphism $$\chi: K_0(X) \to \mathbb{Z}[t]$$.
 * 2) There are multiple common notations in use for the hilbert polynomial and all should be exploited since they all do the same thing: take the degree 0 piece of a graded module.
 * 3) Fine, but this could have been easily edited instead of deleted
 * 4) The m is frequently used, but if you really want, this can be changed to a t
 * 5) The reason for focusing on Free resolutions is because the Hilbert polynomial lifts to an object in $$D^b(X)$$. This is the modern-most interpretation and will help the reader when considering more advanced notations, such as derived schemes etc. Also, small typos can be easily corrected...
 * If you edit Wikipedia, under the account of another user, this is an identity usurpation. If you edit Wikipedia from the computer of someone else, you must first logout from Wikipedia. In any case, it is strongly recommended to create a user login and to use it.
 * It is task to fix your mistakes and to harmonize your addition withe the remainder of the article.
 * "Modern-most interpretation": This is your opinion, which is clearly not based on any reliable source. Reader that are interested in scheme theory normally search information on scheme theory articles. The article must be accessible for readers that do not know of scheme theory and derived functors. In any case, every notation must be defined in the article where it is used, unless if it standard in all areas related to the article.
 * Sure that I could rewrite your text to make it conform to Wikipedia standards, but I am not willing to spend time for that. Nevertheless, I have done this by adding the section "Relation with free resolutions". Apparently you have not read it, or if you did, your appear to be unable to realize that it results from a cleaning up of your first edits.
 * D.Lazard (talk) 20:04, 1 December 2017 (UTC)


 * So are you suggesting I put the material for graded rings into the "Relation with free resolutions" section and the rest using coherent sheaves in the last section? I did notice it and that's why I put the examples right afterwards.
 * Username6330 (talk) 02:17, 2 December 2017 (UTC)
 * I do not suggest anything to you except the following: 1/ Learn the style, guidelines and policies of Wikipedia by improving the existing material through minor edits and clarifications. 2/ If you find that some article requires some new material, or if you are reverted, do not edit further the article. Instead, discuss on the talk until a WP:CONSENSUS is found.
 * The reason for these suggested limitations is that it is clear, for the moment, that you are unable of inserting new material in a way that respects style, guidelines and policies of Wikipedia.
 * By the way, there is stronger reason for which your examples are not useful as their are (and for which free resolutions are never a good method for computing Hilbert series and Hilbert polynomial). Free resolutions are not only difficult to compute, but they provide also a form of the Hilbert polynomial that cannot be used directly. In fact it requires further non-elementary combinatorial computations for knowing the degree (dimension of the variety) and the leading coefficient (degree of the variety) of the Hilbert polynomial. D.Lazard (talk) 17:16, 2 December 2017 (UTC)


 * Ah, but I am discussing adding new material onto this page. Your dependence on grobner bases is bad because it removes the intuition behind the Hilbert polynomial and its applications in algebraic geometry (which is where it is mostly used). There is no reason that these examples using free resolutions are bad since they are easily deduced in the examples I gave (which are non-complete intersections). I am not advocating the POV that they are the only method for doing so, but do give a tool for coming up with non-trivial examples **by hand**. Also, Wikipedia follows a "be bold" policy for edits and adding new content. The amount of useful mathematical material on this website is disgustingly low. Promoting such a conservative viewpoint about what material enters into the mathematics portion of wikipedia is destructive IMO. I'm sorry you are so focused on promoting Grobner bases, but I do not know a single person that relies on computing hilbert polynomials, by hand, using a grobner basis. If you still remain so insistent, then you are effectively stating that mathematicians must be dependent on a computer for doing their calculations. — Preceding unsigned comment added by 128.138.65.80 (talk) 23:03, 4 December 2017 (UTC)

Proposed Updates
I am using this space to get an okay for changes to the free resolution section. — Preceding unsigned comment added by 73.181.114.81 (talk • contribs) 02:36, 11 December 2017 (UTC)

Relation with free resolutions
Every graded module $M$ over a graded regular ring $R$ has a graded free resolution, meaning there exists an exact sequence
 * $$ 0 \to L_k \to \cdots \to L_1 \to M \to 0,$$

where the $$L_i$$ are graded free modules, and the arrows are graded linear maps of degree zero.

The additivity of Hilbert series implies that
 * $$HS_M(t) =\sum_{i=1}^k (-1)^{i-1}HS_{L_i}(t).$$

If $$R=k[x_1, \ldots, x_n]$$ is a polynomial ring, and if one knows the degrees of the basis elements of the $$L_i,$$ then the formulas of the preceding sections allow deducing $$HS_M(t)$$ from $$HS_R(t) = 1/(1-t)^n.$$ In fact, these formulas imply that, if a graded free module $L$ has a basis of $h$ homogeneous elements of degrees $$\delta_1, \ldots, \delta_h,$$ then its Hilbert series is
 * $$HS_L(t) = \frac{t^{\delta_1}+\cdots +t^{\delta_h}}{(1-t)^n}.$$

These formulas may be viewed as a way for computing Hilbert series. This is rarely the case, as, with the known algorithms, the computation of the Hilbert series and the computation of a free resolution start from the same Gröbner basis, from which the Hilbert series may be directly computed with a computational complexity which is not higher than that the complexity of the computation of the free resolution.

Simple Quotient Algebra
Consider the non-complete intersection $$R=\mathbb{C}[x,y,z]$$-algebra $$R/(xy,xz) = R/I$$. There is a graded-free resolution of $$R/I$$ as an $$R$$-module given by

0 \to R(-3) \xrightarrow{ \begin{bmatrix} z \\ -y \end{bmatrix} } R(-2) \oplus R(-2) \xrightarrow{ \begin{bmatrix} xy & xz \end{bmatrix} } R \to R/(xy,xz) \to 0 $$ Using this we can compute the Hilbert polynomial as

HP_{R/I}(t) = HP_R(t) - 2HP_{R(-2)}(t) + HP_{R(-3)}(t) $$ For $$t$$ large enough we can compute the hilbert polynomial using binomial coefficients

HP_{R(-a)}(t) = {2 + t - a \choose t- a} = \frac{(t + 2 - a)(t + 1 - a)}{2} = \frac{t^2 + (3 - 2a)t + (a^2 - 3a + 2)}{2} $$ So in each case this is

\begin{align} HP_R(t) &= \frac{t^2 + 3t + 2}{2} \\ HP_{R(-2)}(t) &= \frac{t^2 - t}{2} \\ HP_{R(-3)}(t) &= \frac{t^2 - 3t + 2}{2} \end{align} $$


 * I have collapsed the text proposed by }}, for distinguishing this text for the discussion about it.
 * At first glance, this text consists of the verbatim of the section "Relation with free resolutions" of the article, followed by a new subsection. If there are proposed changes into the existing section, they must be made explicit, as the readers of this discussion are not supposed to spent time for checking differences.
 * As it is, the proposed subsection is more confusing than useful for a reader that is not an expert of the subject, because of the following issues.
 * Use, without need and without link or reference of a terminology that may be unknown to the reader (complete intersection)
 * Use of notations that is not defined in the article and may be unknown to the reader ($$R(-a)$$, $$>>$$)
 * Starting by an assertion ("This is a free resolution") that is not proved nor sourced. This is highly confusing, as the first question most people will ask is "how can I get this?)
 * $$HP_R(t) = 1$$ is wrong
 * Although this is supposed to show how computing $$H_{R/I}(t)$$, the result is not given
 * For coherency with the remaining of the article, the Hilbert series should also be computed
 * For being not confusing, it should be explained why using technical tools, when the direct computation is very easy: A basis of $$R/I$$ consists of the powers of $x$ and the monomials in $y, z$. Thus $$HF(0)=1,$$ and $$HF(t)=t+2$$ for $$t > 0.$$ Thus $$HP(t)=t+2$$ and $$HS(t)=1/(1-t) + 1/(1-t)^2-1 =(1+t-t^2)/(1-t)^2.$$
 * More important: examples should be added to the article but they must not be focused to free resolutions (which are not the subject of the article). They must be well chosen for showing how the concepts that are considered here (including free resolutions) are interrelated. More thoughts are needed for well choosing the example(s), and deciding how they should be presented for best illustrating the article. D.Lazard (talk) 08:49, 11 December 2017 (UTC)
 * What terminology should be used for the algebra in this section? You mention the algebra for a complete intersection variety earlier, and that could be used as a definition for a complete intersection algebra, so I don't see the problem.
 * I'll add the twisting notation to part of the page discussing graded algebras
 * Maybe I could include this example of a free resolution on the resolutions page and explain why it is a free resolution there.
 * — Preceding unsigned comment added by 74.220.45.91 (talk • contribs) 03:06, 12 December 2017 (UTC)
 * Please sign your posts in talk page with four tildes ( ~ ). I an tired of doing this for you. D.Lazard (talk) 21:48, 12 December 2017 (UTC)

Degree of a projective variety and Bézout's theorem
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 $R$. However, the Krull dimension of $R$ 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)
 * Good point, at this step, the denominator is still $1 – t$. I'll try fixing this (the result is definitively correct). D.Lazard (talk) 14:14, 6 January 2018 (UTC)
 * 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)