The multigraded Poincare-Betti series P_R^k(x_1,...,x_n; t) of a monomial
ring k[x_1,...,x_n]/ on a finite number of monomial generators has the form
(1+tx_1)(1+tx_2)...(1+tx_n)/b_(R,k)(x_1,...,x_n; t), where
b_(R,k)(x_1,...,x_n;t) is a polynomial depending only on the monomial set M and
the characteristic of the field k. I present a computer program designed to
calculate the polynomial b_(R,k) for a given field characteristic and a given
set of monomial generators.Comment: 8 pages, prepared for the School and Workshop on Algebraic geometry
  and statistics at Politecnico Torino in September 200

Johansson, Mikael

English

arXiv

The multigraded Poincaré-Betti series
$P^kR(\bar{x}; t) of a
monomial ring $k[\bar{x}]/\langle M\rangle$
on a finite number of monomial generators has the form
$\prod_{x_i\in\bar{x}}(1+x_i t)/b_{R,k}(\bar{x};t)$,
where $b_{R,k}(\bar{x};t)$  is a
polynomial depending only on the monomial set M and the
characteristic of the field k. I present a computer program designed to
calculate the polynomial $b_{R,k} for a given field characteristic and
a given set of monomial generators

OpenstarTs

Rend. Istit. Mat. Univ. TriesteVol. XXXVII, 85–94 (2005)Computation of Poincaré-Betti Seriesfor Monomial RingsMikael Johansson (∗)Contribution to “School (and Workshop) on Computational Algebrafor Algebraic Geometry and Statistics”, Torino, September 2004.Summary. - The multigraded Poincaré-Betti series P kR(x̄; t) of amonomial ring k[x̄]/〈M〉 on a finite number of monomial gener-ators has the form∏xi∈x̄(1+xit)/bR,k(x̄; t), where bR,k(x̄; t) is apolynomial depending only on the monomial set M and the char-acteristic of the field k. I present a computer program designed tocalculate the polynomial bR,k for a given field characteristic anda given set of monomial generators.1. IntroductionLet Q = k[x̄] = k[x1, . . . , xr] be the polynomial ring over a fieldk with r variables. The ring has a natural Nr-grading1 by set-ting deg(xi) = ei for the canonical basis vectors ei of Nr. Setting|(a1, . . . , ar)| = a1 + · · · + ar, we can derive an N-grading of Q fromthis Nr-grading. We write deg(m) for the Nr-degree of a monomialm, and given (a1, . . . , ar) = α ∈ Nr, we write xα = xa11 xa22 . . . xarr(∗) Author’s address: Mikael Johansson, Friedrich-Schiller-Univesität Jena, Fac-ultät für Mathematik & Informatik, Matematisches Insitut, D-07737 Jena, e-mail:mik@math.su.seKeywords: Poincaré-Betti series, Monomial rings, Simplicial homology, Computa-tion.AMS Subject Classification: 13D07, 13-04, 13P99.1Throughout this paper, 0 ∈ N.86 M. JOHANSSONThe Nr-grading and its inherent N-grading both are inheritedfrom Q to the ring R = Q/I where I is a monomial ideal, i.e. anideal generated by monomials in Q. We call R a monomial ring.The Nr-grading inherits, via minimal resolutions respecting tothe grading, to an Nr-grading on TorR(k, k). Thus, we can definethe multigraded Poincaré-Betti series PRk (x̄; t) of an Nr-graded ringR over k: ∑i∈N∑α∈Nrdimk TorRi (k, k)αxαtiFrom PRk (x̄; t), the simple Poincaré-Betti series can be calculatedas PRk (t) = PRk (1, . . . , 1; t).Since [3], it is known that the Poincaré-Betti series of a monomialring R holds the form PRk (t) =(1+t)nbR,k(t)for some polynomial bR,k(t).Moreover, Alexander Berglund proved in [4] that deg bR,k(t) < 2nfor monomial rings with n monomial generators. It follows that thereare finitely many Poincaré-Betti serie occurring for a fixed n at all.We may define a partially ordered graph, or po-graph, to be agraph with a partial order on the vertices. Two po-graphs are saidto be isomorphic if there is a simultaneous isomorphism of the graphand the partial order.Let Q = k[x̄] and Q′ = k[x̄′] be polynomial rings for two finitevariable sets x̄ = {x1, . . . , xr} and x̄′ = {x1, . . . , xr′}, and let I andI ′ be monomial ideals in Q and Q′ respectively, with M and M ′the sets of generators for each ideal. For some set S of monomials,we denote by LS the set of all least common multiples of subsets ofS. LS can be equipped with the structure of a po-graph, orderingmonomials by divisibility and adding an edge between two elementswhen they have a non-trivial common factor.Luchezar Avramov shows in [2] that if two rings R = Q/I andR = Q′/I ′, with I and I ′ generated by the monomial sets M and M ′respectively are such that LM ∼= LM ′ , then bR,k(t) = bR′,k(t). Fromthis follows that for a fixed field and a fixed number of monomialgenerators, only finitely many different Poincaré-Betti series can oc-cur. Avramov further proves that the limitation to a fixed field issuperfluous.In [4], Alexander Berglund, proving the conjecture by Avramovthat deg(bR,k(t)) ≤ 2n whenever the monomial ideal generating theCOMPUTATION OF POINCARÉ-BETTI SERIES 87ring R has n generator, constructs combinatorially a minimal modelfor R and gives a characterisation of the Poincaré-Betti series denom-inator polynomial in terms of the homology of associated simplicialcomplexes. Avramov’s earlier observation that only finitely manydifferent Poincaré-Betti series exist for a fixed number of generatorsfor the monomial ideal I follows as an immediate consequence ofBerglund’s construction.I have in the course of my M.Sc. thesis work [7] continued Berg-lund’s work by implementing his formula in a program, written inC++, capable of calculating simplicial homology over fields of arbri-trary characteristic, as well as explicitly calculating bR,k(t) for arbri-trary characteristic of the coefficient field k.1.1. Simplicial ComplexesSince we will work a lot with simplicial homology, I will take a fewmoments to review definitions and terminology. A simplicial complexon a set V is a set ∆ of subsets of V such that if G ∈ ∆ and F ⊂ G,then F ∈ ∆. V is called the vertex set of ∆. All simplicial complexesI shall refer to will have V =⋃∆ unless otherwise stated. The i-faces or i-simplices of ∆ are precisely the elements in ∆ of cardinalityi + 1.To a simplicial complex ∆ we can associate an augmented chaincomplex C̃(∆) with C̃i(∆) the free abelian group on the i-faces of ∆,where we consider ∅ to be the unique −1-simplex. We equip C̃(∆)with the standard differential of degree −1. ThusHi(C̃(∆)) = H̃i(∆)As usual, for an abelian group G, we set C̃(∆;G) = C̃(∆) ⊗Z Gand H̃i(∆;G) = Hi(C̃(∆;G)).For a simplicial complex ∆ we define the Alexander dual∆∨ = {F ⊆ V | V \ F 6∈ ∆}For simplicial complexes ∆ and ∆′ with disjoint vertex sets wedefine the join∆ ∗ ∆′ = {F ∪ F ′ | F ∈ ∆, F ′ ∈ ∆′}88 M. JOHANSSONand the dual join∆ · ∆′ = (∆∨ ∗ ∆′∨)∨and note that then∆ · ∆′ = {F ∈ V ∪ V ′ | F ∩ V ∈ ∆ or F ∩ V ′ ∈ ∆′}In [5, Lemma 5.5.3], it’s shown thatH̃i(∆; k) ∼= H̃n−3−i(∆∨; k)for a complex ∆ on n vertices.For a graded vector space H =⊕i∈N Hi, we will write H(t) forthe generating function∑i∈N dimk Hiti of H. We then can find thatthe join of complexes as well as the Alexander dual gives rise to rathereasily handled equalities on the level of generating functions for theirrespective homologies, again with ∆ a complex on n vertices, and ∆′some other complex.tnH̃(∆∨; k)(t−1) = t3H̃(∆; k)(t) (1.1)H̃(∆ ∗ ∆′; k)(t) = tH̃(∆; k)(t) · H̃(∆′; k)(t) (1.2)where the t factor in the latter equation comes from the fact that asimplex with d elements is considered to have dimension d − 1.2. Berglund’s workIn [4], Berglund treats the theoretical aspects of computation of thePoincaré-Betti series denominator for monomial rings. I will notrepeat all of his proof here, but rather reference his work to establishthe vocabulary and touch the results I will need for my own work.Q = k[x̄] is Nr-graded by assigning to a monomial xa11 . . . xarr theelement α = (a1, . . . , ar). We write xα for xa11 . . . xarr . The monomialxα is said to be squarefree if α ∈ {0, 1}r .By a construction by Weyman and Fröberg [8, 6], it is enoughto treat squarefree monomial sets, since an easy procedure can beused to go from a monomial ring to a squarefree monomial ring withthe same homological properties. So we can assume that R = Q/ICOMPUTATION OF POINCARÉ-BETTI SERIES 89is a squarefree monomial ring with I generated by the monomialantichain2 M of cardinality n.For a set S of monomials, set mS = lcm(m : m ∈ S). In particu-lar m∅ = 1. Thus LM = {mS | S ⊂ M}. For a specific monomial mand a monomial set M set Mm = {m′ ∈ M | m′|m}.Now, for I = 〈M〉 a monomial ideal in Q generated minimallyby the antichain M , Berglund introduces the complex∆M = {S ⊆ M | mS 6= mM or S disconnected}where connectivity is for S as a subgraph of LM .Using multigraded ring-deviationsǫi,α = dimk H̃i−3(∆Mxα ; k)Berglund gives the squarefree part of the multigraded Poincaré-BettipolynomialbR,k(x̄, t) ≡∏xα∈LM(1 − xαpα(t)) (mod 〈x21, . . . , x2r〉) (2.1)with pα(t) = t3H̃(∆Mxα ; k)(t). Backelin demonstrated already in [3]that the denominator polynomial will be squarefree whenever themonomials generating the ideal all are.Berglund then goes on to find several more theoretically pliableforms of this particular formula; expanding the product and takingthe irrelevance of non-squarefree terms into account, he arrives atthe formbR,k(x̄; t) = 1 +∑SmS(−t)c(S)+2H̃(∆S ; k)(t) (2.2)where the sum is taken over all non-empty saturated subsets of Mand c(S) counts the number of graph components of S as a subgraphto LM . We define the saturation of a subset S ⊂ M as the set ofall monomials in M that divide the least common multiple of someconnected component of S as subgraph to LM . A set is saturated ifit is equal to its saturation.2Recall that in a partially ordered set P , an antichain A is a subset suchthat all elements of A are mutually incomparable. The monomials are partiallyordered by divisibility.90 M. JOHANSSON3. The resulting application – poincareMy own achievement is that I have constructed a computer programto calculate simplical complex homology and mainly to calculate thedenominator polynomials of Poincaré-Betti series using Berglundsmethods. I will devote this section to a discussion of the program,which can be fetched in its latest version under the MIT softwarelicense from http://www.math.su.se/~mik/poincare/.The form deemed most promising for implementation as I startedwas the form given in (2.1) – mainly since the formulation in termsof saturated subsets had at that time not yet matured. Thus, Ihave implemented specific C++ classes for calculating in the ringQ/〈x21, x22, . . . , x2r〉 and let the final product forming the polynomialtake place in that particular ring. The only part forming any kind ofcomplexity for the straightforward implementation is that of formingthe complex ∆Mxα and calculating its homology over the specifiedcharacteristic.The construction of ∆Mxα is done with a modified kind of breadth-first search: monomials are stored in a queue along with an in-dex keeping track of which of the monomials covering the particularmonomial that have already been tried. Thus, for each monomial inthe queue, all later covering monomials are tried one after the other,and upon compliance with the two conditions – that the least com-mon multiple of all generating monomials dividing the candidate isequal to xα and that those generating monomials are connected as agraph – the monomial is added to the queue carrying a testing indexone higher than the index that produced it. This algorithm doesyield a speed increase compared to the earlier algorithm that simplytested all monomials for both conditions; but still is not optimal byfar.Once the simplicial complex as such has been constructed, thecalculation of its homology commences. This is calculated degree bydegree, constructing a matrix with entries in {0,±1} and fetching itsrank from the matrix routines in the Pari library [1].This is wrapped in a text-mode user interface, using the GNUReadline library to facilitate command history and command edit-ing. The user interface reads in space-separated lists of monomials asCOMPUTATION OF POINCARÉ-BETTI SERIES 91input to the add simplex and add monomial commands. A mono-mial, to the program, is a *-separated list of strings of characters,where each separated string is taken to be the name of a variable.The variable names must avoid +-*/^,. and whitespace, but can useany other characters. Any string occuring in such a position will beinterpreted as a variable and added to an internal dynamic variablepool.The user interface wraps, among other things, around the Wey-man-Fröberg method for conversion to a squarefree monomial ring.The conversion is done transparently, using several internal variablesthat are easily converted back to the original variables before printingthe answer. The output rendered by the program is written in sucha way that other computer algebra systems should have an easy timehandling it.As an example on how the program works, I give in example 3.1a session, calculating first the simplicial homology of the projectiveplane over Q as well as over Z2 and then calculating the Poincaré-Betti denominator polynomial of Q[x, y, z]/〈x2, xy, yz〉.Example 3.1.Welcome to the Poincaré calculator.You can use this program to calculate simplicialhomology over prime fields and to calculate thedenominator polynomial of the Poincaré-Betti seriesof monomial rings.(c) 2004 Mikael JohanssonThis program is released under the MIT License> add simplex a*b*e a*b*f a*c*d a*c*f a*d*e> add simplex b*c*d b*c*e b*d*f c*e*f d*e*f> homologyCalculating homology ranks...***** Hilbert series of simplicial homology *****0> char 2New characteristic: 2> homologyCalculating homology ranks...***** Hilbert series of simplicial homology *****ZZ + ZZ^2> add monomial x^2 x*y y*z92 M. JOHANSSON> char 0New characteristic: 0> denom1 - x^2*ZZ^2 - x*y*ZZ^2 - y*z*ZZ^2 - x^2*y*ZZ^3 - x*y*z*ZZ^3> set multigrade false> denom1 - 3*ZZ^2 - 2*ZZ^3> quitThanks for visiting.Calculation with poincareAmong the things we may observe in example 3.1 is the basicset of commands – add simplex and add monomial to build simpli-cial complexes or monomial ideals, homology and denominator (oran abbreviation denom thereof) to calculate simplicial homology andthe Poincaré-Betti denominator respectively, as well as the commandchar, which changes the field characteristic over which all homologycalculations take place and the command sequence set multigradefalse, which sets a flag that causes the program to change the wayit prints the polynomials output by the denominator command, sothat instead of the polynomial bR,k(x̄, t) the program prints the poly-nomial bR,k(1, . . . , 1, t). The program ends upon receiving quit.In addition to these, there are the commands clear: clearing thestored simplicial complex and monomial ideal, but not changing thecharacteristics used and var: which changes the implicit homologyvariable, which in my review in this paper has been called t, andwhich by default in poincare is called ZZ. Should you wish to useZZ as a ring variable in your calculations, a change of homologyvariable will be necessary. For this, the command var is provided,with which you can change the string that poincare uses for thehomology variable.4. Questions and future directionsThere are several things that I want to improve upon on the systemherein presented, and also several questions that can be posed.There are numerous complexity issues associated to the programin its current form. Mainly, these issues are related to the size ofCOMPUTATION OF POINCARÉ-BETTI SERIES 93the resulting po-graphs for larger sets of monomials. As an exam-ple, the initial ideal of a Gröbner basis with revlex ordering of thehomogenized cyclic 6-root ideal, i.e. the ideal generated byx1 + x2 + x3 + x4 + x5 + x6,x1x2 + x2x3 + x3x4 + x4x5 + x5x6 + x1x6,x1x2x3 + x2x3x4 + x3x4x5 + x4x5x6 + x1x5x6 + x1x2x6,x1x2x3x4 + x2x3x4x5 + x3x4x5x6 + x1x4x5x6 + x1x2x5x6++ x1x2x3x6,x1x2x3x4x5 + x2x3x4x5x6 + x1x3x4x5x6++ x1x2x4x5x6 + x1x2x3x5x6 + x1x2x3x4x6,x1x2x3x4x5x6 − y6invariably becomes larger than Pari’s working memory. This initialideal has 100 monomial generators and produces a po-graph with11443 elements. The calculations normally halt after between 200and 400 lattice point calculations.The problem I have observed with for instance this example isthat the calculation of homology of large simplicial complexes ismemorywise unfeasible. The most visible problem is when the ho-mology calculations turn out to be too hard, since this results ina crash in the Pari library; whereas too hard construction of thesimplicial complexes merely result in slow running of the program.Question 4.1. Given the rather special structure of the matricesthat are used to calculate field homology of a simplicial complex, cananything be said about the sizes of elements of a matrix as some sortof reduction algorithm is used to deduce its rank? Can some variantof row-reduction be found such that the matrices that produce “large”entries (for instance, larger than a standard 32-bit word) have someeasily recognizable feature?If such a method could be found, then it would be possible tosingle out the specific homology matrices that would actually needtreatment with some sort of bignum library, and produce fast 32-bit(sparse) matrix arithmetic to deal with all other instances. Such aseparate treatment would also obliterate the need to rely on Pari’s94 M. JOHANSSONinternal memory allocation structures and would give the user morecontrol over acceptable memory consumption for the calculations athand.An obvious further point of attack is the generation algorithmfor the Berglund complexes. This would be vastly improved if asearch algorithm would be constructed that minimizes the numberof multiple checks done on each candidate monomial; since the graphconnectivity checks are not, in the context, very fast.Finally, an interesting direction to take would be to look at theAPIs for larger computer algebra systems and try to adapt the codehere written to work as a pluggable module to those systems; forinstance providing an interface to calculate Poincaré-Betti series ofmonomial rings directly from Singular or Magma or Macaulay 2.References[1] Pari/gp, version 2.2.7α, Tech. report, The PARI Group, Bordeaux,2004, available from http://pari.math.u-bordeaux.fr/.[2] L. L. Avramov, Homotopy Lie algebras and Poincaré series of al-gebras with monomial relations, Homology Homotopy Appl. 4 (2002),no. 2, 17–27.[3] J. Backelin, Les anneaux locaux à relations monomiales ont des sériesde Poincaré-Betti rationelles, C. R. Acad. Science Paris 295 (1982),607–610.[4] A. Berglund, Poincaré series of monomial rings, arXiv:math.AC0412282, 2004.[5] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Stud-ies in Advanced Mathematics, vol. 39, Cambridge University Press,1993, revised edition.[6] R. Fröberg, A study of graded extremal rings and of monomial rings,Math. Scand. 51 (1982), 22–34.[7] M. Johansson, Computation of denominator polynomials forPoincaré series on monomial rings, Master’s thesis, Department ofMathematics, Stockholm University, SE-10691 Stockholm, Sweden,2004.[8] J. Weyman, Idea ly generowane przez jednomiany, Master’s thesis,Nicholas Copernicus University, Torun, Poland, 1977.Received December 6, 2005.

Computation of Poincaré-Betti Series for Monomial Rings

Academic Archive On-line (KTH, Sweden)

Computation of Poincare-Betti series for monomial rings

Publikationer från KTH

Digitala Vetenskapliga Arkivet - Academic Archive On-line

https://www.openstarts.units.it/bitstreams/df1b6957-0e35-426e-9afe-90e668166e8b/download

Computation of Poincare-Betti series for monomial rings

Abstract

Similar works

Full text

Available Versions

OpenstarTs

Academic Archive On-line (KTH, Sweden)

Publikationer från KTH

Digitala Vetenskapliga Arkivet - Academic Archive On-line