The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a
method to compute all saturated strongly stable ideals in a given polynomial
ring with a fixed Hilbert polynomial. A description of the main method and
auxiliary tools is given.Comment: Source code available as an ancillary file. Final versio

Alberelli, Davide

Lella, Paolo

English

arXiv

The StronglyStableIdeals.m2 package for Macaulay2 provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary tools is given

Alberelli, D.

Lella, P.

Archivio istituzionale della ricerca - Politecnico di Milano

Journal of Software forAlgebra and GeometryStrongly stable ideals and Hilbert polynomialsDAVIDE ALBERELLI AND PAOLO LELLAvol 9 2019JSAG 9 (2019), 1–9 The Journal of Software fordx.doi.org/10.2140/jsag.2019.9.1 Algebra and GeometryStrongly stable ideals and Hilbert polynomialsDAVIDE ALBERELLI AND PAOLO LELLAABSTRACT: The StronglyStableIdeals.m2 package for Macaulay2 providesa method to compute all saturated strongly stable ideals in a given polynomial ringwith a fixed Hilbert polynomial. A description of the main method and auxiliarytools is given.INTRODUCTION. Strongly stable ideals are a key tool in commutative algebraand algebraic geometry. These ideals have nice combinatorial properties that makethem well suited for both theoretical and computational applications. In the caseof polynomial rings with coefficients in a field of characteristic zero, the notionof strongly stable ideals coincides with the notion of Borel-fixed ideals. Suchideals are fixed by the action of the Borel subgroup of triangular matrices andplay a special role in the theory of Gröbner bases because initial ideals in genericcoordinates are of this type [Galligo 1974].In the context of parameter spaces of algebraic varieties, Galligo’s theorem saysthat each component and each intersection of components of a Hilbert schemecontains at least one point corresponding to a scheme defined by a Borel-fixedideal. Hence, these ideals are distributed throughout the Hilbert scheme and can beused to study its local structure. To this end, in recent years several authors [Lellaand Roggero 2011; 2016; Cioffi and Roggero 2011; Bertone et al. 2013a; 2017a;2017b] developed algorithmic methods based on the use of strongly stable idealsto construct flat families corresponding to special loci of the Hilbert scheme. Inparticular, a new open cover of the Hilbert scheme has been defined using stronglystable ideals and the action of the projective linear group [Bertone et al. 2013b;Brachat et al. 2016]. In this construction, the list of all points corresponding toBorel-fixed ideals in a given Hilbert scheme is needed. The main feature of thepackage StronglyStableIdeals.m2 is a method to compute this set of points,i.e., the list of all saturated strongly stable ideals in a polynomial ring with a givenThe second author is a member of GNSAGA .MSC2010: primary 13P10; secondary 13P99.Keywords: strongly stable ideal, Borel-fixed ideal, Hilbert polynomial, Gotzmann number, Hilbertscheme.StronglyStableIdeals.m2 version 1.112 Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomialsHilbert polynomial. The method has been theoretically introduced in [Cioffi et al.2011] and improved in [Lella 2012]. Several other tools are developed and pre-sented in the current paper.1. STRONGLY STABLE IDEALS. Let us denote by K[x] the polynomial ring inn + 1 variables K[x0, . . . , xn] with coefficients in a field K. We assume thatx0 > x1 > · · ·> xn . We use the multi-index notation to describe monomials, i.e.,xα := xα00 · · · xαnn for every α = (α0, . . . , αn) ∈ Zn+1>0 , and we denote by Tn,s the setof monomials of K[x] of degree s. For any monomial xα, we denote by min xαand max xα the indices of the minimal and maximal variable dividing xα.Following [Green 2010], increasing and decreasing elementary moves are de-fined as the multiplicationse+i (xα) := xi−1xi· xα, i > 0, and e−j (xα) :=x j+1x j· xα, j < n.We say that an elementary move e+/−i is admissible for a monomial xα if αi > 0,that is, e+/−i (xα) is a monomial of K[x].Definition 1.1. An ideal I ⊂ K[x] is called strongly stable if(i) I is a monomial ideal;(ii) for every xα ∈ I and for every admissible increasing move e+i , the monomiale+i (xα) is contained in I.We recall that a strongly stable ideal is a Borel-fixed ideal. We now summarizesome properties holding in general for Borel-fixed ideals and useful in this context.Proposition 1.2 [Green 2010, Section 2]. Let I ⊂ K[x] be a strongly stable ideal.(i) The regularity of I is equal to the maximal degree of a generator.(ii) Let m be the irrelevant ideal of K[x]. Then, (I : m) = (I : xn), so that theideal I is saturated if no generator involves the last variable xn .(iii) The last variable xn is a regular element for I, i.e., the multiplication by xninduces the short exact sequence0 −→ K[x]I(t − 1) ·xn−→ K[x]I(t) −→ K[x](xn, I )(t) −→ 0.2. HILBERT POLYNOMIALS. The Hilbert polynomial p(t) of a homogeneousideal I ⊂ K[x] is the numerical polynomial such that for s sufficiently largedimK(K[x]I)s= dimK(K[x]sIs)=(n+ sn)− dimK Is = p(s).Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomials 3Obviously, not every numerical polynomial is a Hilbert polynomial of some homo-geneous ideal. Those being Hilbert polynomials have been completely describedby Gotzmann [1978].Gotzmann’s decomposition. A numerical polynomial p(t) ∈ Q[t] is a Hilbertpolynomial if , and only if , it can be written asp(t)=(n+a1a1)+(n+a2−1a2)+· · ·+(n+ar−(r−1)ar), a1> · · ·>ar >0. (1)This decomposition is strictly related to Macaulay’s decompositionp(t)=d∑k=0[( t+kk+1)−( t+k−mkk+1)],where d = deg p(t). For all n > d + 1 the saturated lexicographic ideal withinK[x0, . . . , xn] with Hilbert polynomial p(t) is(x0, . . . , xn−d−2, xbd+1n−d−1, xbdn−d−1xbd−1+1n−d , . . . , xbdn−d−1xbd−1n−d · · · xb0n−1),wherebd = #{a j | a j = d} = md and bk = #{a j | a j = k} = mk −mk+1, 06 k < d.The description of the lexicographic ideal in terms of Gotzmann’s decomposi-tion gives an insight to the following theorem.Gotzmann’s regularity theorem. The regularity of a saturated ideal I ⊂ K[x]with Hilbert polynomial p(t) is at most r , where r is the number of terms in thedecomposition (1) and it is called the Gotzmann number of p(t).Example 2.1. The package StronglyStableIdeals.m2 provides the methodisHilbertPolynomial to determine if a numerical polynomial is a Hilbert poly-nomial.Macaulay2, version 1.11with packages: ConwayPolynomials, Elimination, IntegralClosure,InverseSystems, LLLBases, PrimaryDecomposition,ReesAlgebra, TangentConei1 : loadPackage "StronglyStableIdeals";i2 : QQ[t];i3 : isHilbertPolynomial (4*t)o3 = truei4 : isHilbertPolynomial (5*t-6)o4 = falseGotzmann’s and Macaulay’s decompositions of a Hilbert polynomial can be com-puted using gotzmannDecomposition and macaulayDecomposition. These4 Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomialsmethods return the list of terms in the decompositions. The summand(t+ec)isconstructed with the command projectiveHilbertPolynomial(c,c-e).i5 : gotzmannDecomposition (4*t)o5 = {P , - P + P , - 2*P + P , - 3*P + P , P , P }1 0 1 0 1 0 1 0 0o5 : Listi6 : macaulayDecomposition (4*t)o6 = {- P + P , 7*P - P , - P + P , - 10*P + 5*P - P }0 1 0 1 1 2 0 1 2o6 : ListFinally, the saturated lexicographic ideal L with Hilbert polynomial p(t) in thepolynomial ring K[x] can be computed with the method lexIdeal and its regu-larity is equal to the Gotzmann number of p(t).i7 : L = lexIdeal (4*t, QQ[x,y,z,w])5 4 2o7 = ideal (x, y , y z )o7 : Ideal of QQ[x, y, z, w]i8 : regularity L == gotzmannNumber (4*t)o8 = true3. THE MAIN ALGORITHM. In this section, we outline the strategy of the mainalgorithm. This algorithm was firstly described in [Cioffi et al. 2011] and thenoptimized in [Lella 2012]. The same problem has been previously discussed in[Reeves 1992] and an alternative algorithm was later presented in [Moore andNagel 2014].We need to relate the properties of a strongly stable ideal with its Hilbert poly-nomial. If I is a strongly stable ideal, for each s ∈ N the monomial basis ofthe homogeneous piece Is of the ideal is a subset of Tn,s closed by increasingelementary moves. We call Borel sets such subsets of Tn,s (see Figure 1 for anexample). Proposition 1.2(i) implies that the monomial basis of Is for a saturatedstrongly stable ideal I ⊂K[x] with Hilbert polynomial p(t) and regularity at most sis a Borel set with q(s) := (n+sn )− p(s) elements. Thus, we consider the map{saturated strongly stable ideals in K[x] withHilbert polynomial p(t) and regularity 6 s}↪→{Borel sets of Tn,swith q(s) elements}. (2)Moreover, Gotzmann’s regularity theorem suggests considering s equal to the Gotz-mann number of p(t) to determine all saturated strongly stable ideals with Hilbertpolynomial p(t). Obviously, there are many Borel sets in Tn,s with q(s) elementsnot corresponding to an ideal with Hilbert polynomial p(t). To identify the imageof the previous map, we recall a definition and a proposition by Mall.Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomials 5x30 x20 x1 x0x21 x31x20 x2 x0x1x2 x21 x2x0x22 x1x22x32x40 x30 x1 x20 x21 x0x31 x41x30 x2 x20 x1x2 x0x21 x2 x31 x2x20 x22 x0x1x22 x21 x22x0x32 x1x32x42Figure 1. The Borel sets defined in T2,3 and T2,4 by the ideal(x20 , x0x1, x41)⊂ K[x0, x1, x2].Definition 3.1 [Mall 1997, Definition 2.7]. Let B ⊂ Tn,s be a Borel set. The setB(i) := {xα ∈ B |min xα = n− i} is called the i -growth class of B. The sequencegv(B) := (|B(0)|, . . . , |B(n)|) is called the growth vector of B.Proposition 3.2 [Mall 1997, Proposition 3.2]. Let I ⊂ K[x] be a strongly stableideal generated by the monomials of a Borel set B ⊂ Tn,s and let p(t) be its Hilbertpolynomial. Then,p(t)=(n+tn)−n∑k=0|B(k)|(k+t−sk), for all t > s. (3)We can use this result to determine the growth vector of a Borel set B ⊂ Tn,sstarting from the Hilbert polynomial. The i-th difference polynomial of p(t) is(1i p)(t)= (1i−1 p)(t)− (1i−1 p)(t −1)=(n+t−in−i)−n∑k=i|B(k)|(k+t−s−ik−i).Evaluating these identities at t = s, we obtain the linear system∑nk=0|B(k)| =(n+sn)− p(s),...∑nk=i |B(k)| =(n+s−in−i)− (1i p)(s),...|B(n)| = (s0)− (1n p)(s),(4)whose solution is|B(i)|=n∑k=i|B(k)|−n∑k=i+1|B(k)|=(n+s−i−1n−i)−(1i p)(s)+(1i+1 p)(s), i<n,and |B(n)| = 1 (recall that (1i p)(t)≡ 0 for i > deg p(t) and deg p(t) < n). Let uscall the growth vector of p(t) in degree s the solution of the linear system (4) andlet us denote it by gvs(p(t)).6 Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomialsx50 x40 x1 x30 x21 x20 x31 x0x41 x51x40 x2 x30 x1x2 x20 x21 x2 x0x31 x2 x41 x2x30 x22 x20 x1x22 x0x21 x22 x31 x22x20 x32 x0x1x32 x21 x32x0x42 x1x42x52x50 x40 x1 x30 x21 x20 x31 x0x41 x51x40 x2 x30 x1x2 x20 x21 x2 x0x31 x2 x41 x2x30 x22 x20 x1x22 x0x21 x22 x31 x22x20 x32 x0x1x32 x21 x32x0x42 x1x42x52Figure 2. Borel sets in T2,5 corresponding to the saturated stronglystable ideals (x30 , x20x1, x0x41) (on the left) and (x30 , x20x21 , x0x31) (onthe right) in 3 variables with Hilbert polynomial t + 6 and regularityat most 5.Proposition 3.3 (cf. [Lella 2012, Theorem 3.3]). Let p(t) be a Hilbert polynomial.There is a bijective mapsaturated strongly stable idealsin K[x] with Hilbert polynomialp(t) and regularity 6 s 1:1←→Borel sets of Tn,swith q(s) elements andgrowth vector gvs(p(t)) ,I −→ monomial basis of Is,saturation of (B) ←− B.(5)In order to determine the Borel sets of Proposition 3.3, we use a recursive algo-rithm based on Proposition 1.2(iii). Indeed, if I ⊂K[x0, . . . , xn] is a strongly stableideal with Hilbert polynomial p(t) and B is the associated Borel set in Tn,s , thenthe subset B ′ = {xα ∈ B |min xα > n} ⊂ B is a Borel set in Tn−1,s correspondingto the strongly stable ideal I ′ = (xn, I )∩K[x0, . . . , xn−1] ⊂ K[x0, . . . , xn−1] withHilbert polynomial (1p)(t).Example 3.4. We want to determine the set of strongly stable ideals in the poly-nomial ring K[x0, x1, x2] with regularity at most 5 defining schemes with Hilbertpolynomial p(t)= t + 6. The Gotzmann number of p(t) is 6 and its growth vectorin degree 5 is gv5(t + 6)= (5, 4, 1). We start considering the set of strongly stableideals in K[x0, x1] with Hilbert polynomial 1p(t) = 1 and regularity at most 5corresponding to Borel sets with growth vector gv5(1p(t)) = (4, 1). There is aunique Borel setB ′ = {x50 , x40x1, x30x21 , x20x31 , x0x41} .Since x51 is not contained in B′, a Borel set B ⊂ T2,5 with growth vector (5, 4, 1)does not contain monomials obtained from x51 by applying decreasing elementarymoves, i.e., x41x2, x31x22 , x21x32 , x1x42 and x52 . Hence, we need to select five mono-mials divisible by both x0 and x2 producing a set closed by increasing elementarymoves (see Figure 2).Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomials 7Our package provides the method stronglyStableIdeals to compute the setof strongly stable ideals of a given polynomial ring with fixed Hilbert polynomialand bounded regularity.i9 : stronglyStableIdeals (4*t, QQ[x,y,z,w])5 4 2 2 4 5o9 = {ideal (x, y , y z ), ideal (x*z, x*y, x , y z, y ),2 2 4 2 3ideal (x*y, x , x*z , y ), ideal (x*y, x , y )}o9 : Listi10 : stronglyStableIdeals (4*t, QQ[x,y,z,w], MaxRegularity => 4)2 2 4 2 3o10 = {ideal (x*y, x , x*z , y ), ideal (x*y, x , y )}o10 : List4. SEGMENT IDEALS. The transitive closure of the order relationxα >B xβ ⇐⇒ xβ = e−i (xα) (6)induces a partial order on the set of monomials of any degree called the Borel order.Every graded term ordering is a refinement of this partial order. Since a Borel set Bis closed with respect to the Borel order, i.e., xα >B xβ, xβ ∈ B ⇒ xα∈ B, it isnatural to ask whether there exists a term ordering ≺ with the same property. Forinstance, for the lexicographic ideal, the graded lexicographic order separates, ineach degree, monomials contained in the ideal from those outside. In [Cioffi et al.2011], several notions of segment ideals are introduced.Definition 4.1 [Cioffi et al. 2011, Definitions 3.1 and 3.7]. A Borel set B ⊂ Tn,s iscalled a segment if there exists a term ordering ≺ such that xα  xβ, for all xα ∈ Band xβ ∈ Tn,s \ B.Let I ⊂ K[x] be a saturated strongly stable ideal.(i) I is called a hilb-segment if the Borel set I ∩Tn,r is a segment, where r is theGotzmann number of the Hilbert polynomial of I.(ii) I is called a reg-segment if the Borel set I ∩Tn,m is a segment, where m isthe regularity of I.(iii) I is called a gen-segment if there exists a term ordering ≺ such that xα  xβfor each minimal generator xα of degree s of I and for all xβ ∈ Tn,s \ Is .These notions are very important in the construction of flat families based onproperties of Gröbner bases and in general for the study of the Hilbert scheme.The StronglyStableIdeals.m2 package provides three methods for determin-ing whether a strongly stable ideal may be some type of segment (and, in case, givesthe term ordering). These methods use tools of the package gfanInterface.m2and the term ordering is given as a weight vector.8 Alberelli and Lella :::: Strongly stable ideals and Hilbert polynomialsi11 : sevenPointsP2 = stronglyStableIdeals (7, 3, MaxRegularity => 5)2 2 5 2 4 3o11 = {ideal (x , x x , x ), ideal (x , x , x x ),0 0 1 1 0 1 0 12 2 3 4ideal (x x , x x , x , x )}0 1 0 1 0 1o11 : Listi12 : for J in sevenPointsP2 list isHilbSegment Jo12 = {(true, {7, 3, 1}), (false, ), (true, {4, 3, 1})}o12 : Listi13 : for J in sevenPointsP2 list isRegSegment Jo13 = {(true, {7, 3, 1}), (false, ), (true, {4, 3, 1})}o13 : Listi14 : for J in sevenPointsP2 list isGenSegment Jo14 = {(true, {6, 3, 1}), (true, {4, 3, 1}), (true, {4, 3, 1})}o14 : ListSUPPLEMENT. Version 1.1 of StronglyStableIdeals.m2 is contained in theonline supplement.REFERENCES.[Bertone et al. 2013a] C. Bertone, F. Cioffi, P. Lella, and M. Roggero, “Upgraded methods forthe effective computation of marked schemes on a strongly stable ideal”, J. Symbolic Comput. 50(2013), 263–290. MR Zbl[Bertone et al. 2013b] C. Bertone, P. Lella, and M. Roggero, “A Borel open cover of the Hilbertscheme”, J. Symbolic Comput. 53 (2013), 119–135. MR Zbl[Bertone et al. 2017a] C. Bertone, F. Cioffi, and M. Roggero, “Double-generic initial ideal andHilbert scheme”, Ann. Mat. Pura Appl. 196:1 (2017), 19–41. MR Zbl[Bertone et al. 2017b] C. Bertone, F. Cioffi, and M. Roggero, “Macaulay-like marked bases”, J.Algebra Appl. 16:5 (2017), 1750100, 36. MR Zbl[Brachat et al. 2016] J. Brachat, P. Lella, B. Mourrain, and M. Roggero, “Extensors and the Hilbertscheme”, Ann. Sc. Norm. Super. Pisa Cl. Sci. 16:1 (2016), 65–96. MR Zbl[Cioffi and Roggero 2011] F. Cioffi and M. Roggero, “Flat families by strongly stable ideals and ageneralization of Gröbner bases”, J. Symbolic Comput. 46:9 (2011), 1070–1084. MR Zbl[Cioffi et al. 2011] F. Cioffi, P. Lella, M. G. Marinari, and M. Roggero, “Segments and Hilbertschemes of points”, Discrete Math. 311:20 (2011), 2238–2252. MR Zbl[Galligo 1974] A. Galligo, “À propos du théorème de-préparation de Weierstrass”, pp. 543–579in Fonctions de plusieurs variables complexes, edited by F. Norguet, Lecture Notes in Math. 409,Springer, 1974. MR Zbl[Gotzmann 1978] G. Gotzmann, “Eine Bedingung für die Flachheit und das Hilbertpolynom einesgraduierten Ringes”, Math. Z. 158:1 (1978), 61–70. MR Zbl[Green 2010] M. L. Green, “Generic initial ideals”, pp. 119–186 in Six lectures on commutativealgebra, edited by J. Elias et al., Birkhäuser Verlag, Basel, 2010. MRAlberelli and Lella :::: Strongly stable ideals and Hilbert polynomials 9[Lella 2012] P. Lella, “An efficient implementation of the algorithm computing the Borel-fixedpoints of a Hilbert scheme”, pp. 242–248 in ISSAC 2012—Proceedings of the 37th InternationalSymposium on Symbolic and Algebraic Computation, edited by J. van der Hoeven and M. van Hoeij,ACM, New York, 2012. MR Zbl[Lella and Roggero 2011] P. Lella and M. Roggero, “Rational components of Hilbert schemes”,Rend. Semin. Mat. Univ. Padova 126 (2011), 11–45. MR Zbl[Lella and Roggero 2016] P. Lella and M. Roggero, “On the functoriality of marked families”,J. Commut. Algebra 8:3 (2016), 367–410. MR Zbl[Mall 1997] D. Mall, “Betti numbers, Castelnuovo Mumford regularity, and generalisations of Macau-lay’s theorem”, Comm. Algebra 25:12 (1997), 3841–3852. MR Zbl[Moore and Nagel 2014] D. Moore and U. Nagel, “Algorithms for strongly stable ideals”, Math.Comp. 83:289 (2014), 2527–2552. MR Zbl[Reeves 1992] A. A. Reeves, Combinatorial structure on the Hilbert scheme, Ph.D. thesis, Cor-nell University, Ann Arbor, MI, 1992, Available at https://search.proquest.com/docview/303996863.MRRECEIVED: 26 Jun 2014 REVISED: 22 Jun 2018 ACCEPTED: 4 Nov 2018DAVIDE ALBERELLI:davide.alberelli@gmail.comPAOLO LELLA:paolo.lella@polimi.itDipartimento di Matematica, Politecnico di Milano, Milano, Italymsp1JOURNAL OF SOFTWARE FOR ALGEBRA AND GEOMETRY vol 9, no 1, 20191Strongly stable ideals and Hilbert polynomialsDavide Alberelli and Paolo Lella11DiffAlg: a Differential algebra packageManuel Dubinsky, César Massri, Ariel Molinuevo and Federico Quallbrunn19Matroids: a Macaulay2 packageJustin Chen1948-7916(2019)9:1;1-D

Strongly stable ideals and Hilbert polynomials

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Strongly stable ideals and Hilbert polynomials

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