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FINDING SPARSE SYSTEMS OF PARAMETERS

arXiv:alg-geom/9305010v1 20 May 1993FINDING SPARSE SYSTEMS OF PARAMETERSDavid EisenbudBrandeis University, Waltham MA 02254eisenbud@math.brandeis.eduandBernd SturmfelsCornell University, Ithaca, NY 14853bernd@math.cornell.eduAbstract:For several computational procedures such as finding radicals andNoether normalizations, it is important to choose as sparse as possible a systemof parameters in a polynomial ideal or modulo a polynomial ideal. We describenew strategies for these tasks, thus providing solutions to problems (1) and (2)posed in [Eisenbud-Huneke-Vasconcelos 1992].To accomplish the first task we introduce a notion of “setwise complete in-tersection”.

We prove that a set of monomials generating an ideal of codimensionc in a polynomial ring can be partitioned into c disjoint sets forming a setwisecomplete intersection, although the corresponding result is false for arbitrary setsof polynomials. We reduce the general case to the monomial case by a deforma-tion argument.

For homogeneous ideals the output is homogeneous. Our analysisof the second task is based on a concept of Noether complexity for homogeneousideals and its characterization in terms of Chow forms.IntroductionLet k be a field and let S := k[x1, .

. ., xm] be the polynomial ring.

Let J be an idealof S, possibly 0, and let R = S/J.Given a finite set F ⊂R, generating a properideal I, it is a prerequisite for many algebraic computations to find a maximal system ofparameters in the ideal I. By this we mean a system of codim(I) elements of I whichgenerate an ideal of the same codimension; see for example [Eisenbud-Huneke-Vasconcelos1992, Eisenbud 1993, Krick-Logar 1991, and Vasconcelos 1993a,b].

The feasibility of thesubsequent computation often hinges on the fact that the system of parameters is “nice”;typically, that it consists of polynomials which are reasonably sparse, and of low degree.In this paper we address the question of how to compute a system of parameters whichis as sparse as possible. Given a set of polynomials F that generate an ideal of codimensionc, we study the ways of dividing F into subsets F1, F2, .

. ., Fc ⊆F such that if fi is a1

sufficiently general linear combination of elements from Fi then f1, . .

., fc form a systemof parameters modulo J.In the first section we treat the case J = 0. This arises when computing the radicalof an ideal.

Our main result in this case is Theorem 1.3, which says that if the elements ofF are monomials, and somewhat more generally, then the Fi may be chosen to be disjointsubsets of F. We give examples to show that this result fails for arbitrary polynomials,but we can reduce the general case to this one by a deformation argument, using partialGr¨obner bases. Although it is unpleasant to have to compute a Gr¨obner basis for thispurpose, and the worst-case complexity of the computation is certainly very bad, thepayoffis high: if the input polynomials are homogeneous of varying degrees, the outputcan be made homogeneous without any loss of sparseness from the inhomogeneous case.In the second section we consider the case of arbitrary J, and we present a simple“greedy” algorithm (2.1).

We then concentrate on the important case when J is homo-geneous and unmixed, and I is the ideal (x1, . .

., xm). This is the problem of Noethernormalization.In this case we compare some possible meanings of the term “sparse”,with the conclusion that the “correct” measure of sparseness will vary with the applicationat hand.

Choosing one of these, we define the Noether complexity, which is the sparse-ness of the sparsest possible Noether normalization. It is characterized in terms of theChow form of J (Theorem 2.7) and can thus be computed in single-exponential time inm (cf.

[Caniglia 1990]). Another simple approach to computing a Noether normalizationof J is to lift one from any initial ideal of J; this lifting method is explained followingProposition 2.8.

We demonstrate by explicit examples that, in general, neither this liftingmethod nor the greedy algorithm attains the Noether complexity.To simplify the discussion, we assume throughout that k is an infinite field, though inpractice any sufficiently large field will do. In order to use our algorithms it is necessaryto compute the codimensions of various ideals.

Good methods for doing this are discussedin Bayer-Stillman [1992] and Bigatti-Caboara-Robbiano [1993].Both authors were partially supported by the NSF during the preparation of thispaper. The second author was also supported by an A.P.

Sloan Fellowship.2

1. Systems of parameters in a polynomial ringWe retain the notation of the introduction.

In this section we treat the case J = 0, andassume that F is a subset of a proper ideal I of the polynomial ring S := k[x1, . .

., xm].Some obvious approaches and their drawbacksLet c be the codimension of I. A set of c linear combinations of the polynomials in Fwith sufficiently general coefficients in k is a system of parameters, but unfortunately doesnot have the desired sparseness.

Further, if the polynomials in F are homogeneous butof different degrees and a homogeneous system of parameters is required, then in thisapproach one must first replace F by a set of polynomials all of the same degree, forexample by multiplying each one by a power of a generic linear form, or by replacingeach by the ideal it generates in degree equal to the maximal degree in F. This processdramatically destroys sparseness, and raises the degrees of the elements of F in a way thatseems unnecessary.It is thus natural to ask for the smallest subsets F1, F2, . .

., Fc ⊆F such that thelinear combinationsf1 =Xf∈F1r1,f · f ,f2 =Xf∈F2r2,f · f , . .

. ,fc =Xf∈Fcrc,f · f(∗)generate an ideal of codimension c (that is, form a system of parameters) for some choiceof coefficients ri,f.

Supposing that no proper subset of F generates an ideal of codimensionc, an optimal result of this type would be to take F1, F2, . .

., Fc to be a partition of F,that is, disjoint subsets whose union is F.A first hope might be that one could define the sets Fj inductively by the conditionthatF1 ∪F2 ∪. .

. ∪Fjis the smallest initial subset generating an ideal of codimension j.

But this is wrong evenfor monomial ideals as the following example from [Eisenbud 1993] shows:Cautionary Example 1.1. Let m = 4 and F := {x1x2, x2x3, x24, x1x3}.

We have c = 3and the partition suggested above is{x1x2}, {x2x3, x24}, {x1x3}.However, no sequence of the formx1x2 , λx2x3 + µx24 , x1x3is a system of parameters, since it is contained in the height 2 ideal (x1, λx2x3 + µx24). Onthe other hand, the partition{x1x2}, {x2x3, x1x3}, {x24}does have the desired property.3

Unfortunately, partitions with the desired property need not exist.The followingexample was worked out in conversation with Joe Harris.Cautionary Example 1.2.Let {Cij}1≤i≤j≤5 be any 10 distinct irreducible space curvesin P3, and take d an integer large enough so that for each i the ideal of the union of the 6Cpq whose indices p, q do not include i is generated by forms of degree d. For i = 1, . .

., 5let gi be a general form of degree d vanishing on these 6 curves Cpq. Let F = {g1, .

. ., g5}.It is easy to see that F has no zeros in P3 and hence generates an ideal of codimension 4.We claim that there is no partition of F into four disjoint subsets Fi and choice ofcoefficients ri,f such that (∗) generates an ideal of codimension 4.

If such a partitionexisted, then 3 of the sets Fi would have to be singletons. Hence some 3 forms gi, gj, gkwould have to be a system of parameters, and vanish only at finitely many points in P3.Since gi, gj, gk vanish on the curve Cuv, where {i, j, k, u, v} = {1, .

. ., 5}, this is impossible.There is no example of this type with d = 2, but here is one with d = 3: Let p1, .

. ., p5be general points in P3, let Cij be the line passing through pi and pj, and let lijk be theequation of the plane containing the three points pi, pj, pk.

The 6 lines not involving aparticular index i form a tetrahedron. The ideal of their union is generated by the set Fiof 4 cubic forms made by taking products, 3 at a time, of the lstu with i ̸∈{s, t, u}, so wemay take d = 3 in the argument above.Theorem 1.3 below will show that good partitions do exist for monomial ideals, andthis is the basis of our method.

The reason that they exist is essentially that a monomialideal of codimension c is always contained in an ideal generated by c elements — in fact,by c of the variables.A method for finding a sparse system of parametersThe following theorem is our first main result. We fix any term order “≺” on the polynomialring S and we write in≺(F) for the set of initial terms of the polynomials in F.Theorem 1.3.

Suppose that in≺(F) generates an ideal of codimension c. There existpartitions F = F1 ∪. .

. ∪Fc such that for each i the monomials in≺(Fi) have a variablein common.

If F is any such partition, then for almost all ri,f ∈k, the polynomials (∗)generate an ideal of codimension c. Further, each f ∈F may be multiplied by any factorof in≺(f) without spoiling this property.It is known that the hypothesis of Theorem 1.3 is satisfied if F is a Gr¨obner basisof an ideal of codimension c. (See e.g. [Kalkbrener-Sturmfels 1993]).

This suggests thefollowing algorithm for finding a sparse system of parameters.Algorithm 1.3’.Input :A set of generators F for an ideal I of codimension c4

Enlarge F step by step toward a Gr¨obner basis, using the Buchberger algorithm, untilin≺(F) generates an ideal of codimension c. Next replace this partial Gr¨obner basis bya minimal subset which has the same property. Partition this new F into subsets Fi asin Theorem 1.3, for example as follows: Choose a prime (xi1, .

. ., xic) containing in≺(F).Such primes exist because every associated prime of a monomial ideal is generated by asubset of the variables.

For p = 1, 2, . .

., c define Fp inductively to be the set of all elementsof F −∪j≤pFj whose initial terms are divisible by xip.If the polynomials in F are homogeneous, and a homogeneous system of parametersis desired, let di be the maximal degree in Fi, and multiply each polynomial in Fi by apower of one of the variables in its own initial term to bring it up to degree di.Choose random elements ri,f in k, and verify that the polynomials fi in (∗) generatean ideal of codimension c. If they do not, try a new random choice.Output :The sequence f1, . .

., fc.Before starting Algorithm 1.3′, it may be worthwhile to change to the order for whichthe initial forms of F generate an ideal of largest possible codimension. This can be doneusing the polyhedral methods in [Gritzmann-Sturmfels 1993].One subtask to be solved in Algorithm 1.3′ is to find the minimal prime (xi1, .

. ., xic).This is often an easy task, but we remark that in general it amounts to solving a combina-torial problem which is NP-complete.

To see this, consider the case where in≺(F) consistsof square-free quadratic monomials xixj, representing the edges of a graph G with vertexset {x1, . .

., xm}. A subset S of the vertices of G is called stable if no two elements in Sare connected by an edge in G. Our subtask amounts to finding a maximal stable set ofG, a problem which is known to be NP-complete.Our proof of Theorem 1.3 is based on the following criterion for a sequence of sets ofpolynomials to be what one might describe as a “setwise system of parameters”:Proposition 1.4.

Let F1, . .

., Fc ⊂S be sets of polynomials. If for every U ⊆{1, .

. ., c}the set of polynomials Sj∈U Fj generates an ideal of codimension ≥card U, then foralmost every choice of ri,f in k the polynomials f1, .

. ., fc in (∗) generate an ideal ofcodimension c in S.Remarks.

(i) The term “for almost every choice” in Proposition 1.4 (and the term “random” inAlgorithm 1.1) means that ri,f can be chosen in some Zariski-open subset of thecoefficient space. (ii) Example 1.1 shows that we cannot weaken the hypothesis by restricting U to be aninitial subset of {1, .

. ., c}.

(iii) The converse of Proposition 1.4 holds after localizing at any prime containing all the5

Fi’s. (Reason: In a local domain with saturated chain condition, codim (f1, .

. ., fc) =c implies codim (f1, .

. ., fd) = d for all d < c.) It follows that it holds, even withoutlocalizing, if each Fi consists of homogeneous polynomials of positive degree.

(Rea-son: The local codimension of the ideal generated by Sj∈U Fj is minimized in thelocalization at the origin. )The following example shows that the converse is false in general.

We are grateful toAlicia Dickenstein for pointing it out.Cautionary Example 1.5.Let m = 3, F1 = {f1 = (x1+1)x2}, F2 = {f2 = (x1+1)x3},and F3 = {f3 = x1}. Then f1, f2, f3 generate an ideal of codimension 3 but f1, f2 generatean ideal of codimension 1.ProofsProof of Proposition 1.4.

We first show that the conclusion holds in the “generic situation”.Let k′ be the polynomial ring with variables ri,f for f ∈Fi and i = 1, . .

., c. We willshow that the polynomials f1, . .

., fc in (∗) generate an ideal of codimension c in S ⊗k k′.Equivalently, let A be the affine space with the ri,f as coordinates, and let B be the affinespace with coordinates x1, . .

., xm. Let X be the subvariety of A×B defined by f1, .

. ., fc.We will show that the codimension of X is c.Let π2 : X →B be the projection to the second factor of the product.

The fiber ofπ2 over a point p ∈B is a linear space. Since the fi involve disjoint sets of variables ri,f,its codimension equals c minus the number of indices i such that the polynomials in Fiall vanish at p. For each subset U ⊆{1, .

. ., c} let YU denote the set of p ∈B such thatthe polynomials in ∪j∈UFj all vanish at p but some polynomial in each Fj for j /∈U doesnot vanish at p. The constructible sets YU define a stratification of B such that over eachstratum the fibers of π2 have constant codimension c −card U.

Therefore the codimensionof X in A × B equals minc −card U + codim YU : U ⊆{1, . .

., c}. The hypothesis in(c) states that if YU is nonempty, then the codimension of YU is ≥card U.

We concludethat the codimension of X is ≥c.To conclude the proof, consider the projection π1 : X →A to the first factor of theproduct. We must show that the codimension in B of almost every fiber of π1 is ≥c.

Forany dominant map of irreducible varieties, almost every fiber has dimension equal to thedimension of the source minus the dimension of the target. Thus in our situation almostevery fiber has codimension in B equal to the codimension in A × B of the union of thosecomponents of X that dominate A.

This second codimension is ≥the codimension of Xin A × B, which we have shown to be ≥c.Proof of Theorem 1.3.The codimension of the ideal generated by the initial terms of aset of polynomials is ≤that of the ideal of the polynomials themselves. Thus if the sets6

in≺(Fi) satisfy the hypothesis of Proposition 1.4, then so do the sets Fi. Consequentlyit suffices to treat the case where F consists of monomials.We must find a partitionsatisfying the first condition of Theorem 1.3, and show that any such partition satisfiesthe hypothesis of Proposition 1.4.Since every associated prime of a monomial ideal is generated by a subset of thevariables, we may assume (after renumbering variables if necessary) that F is containedin the ideal (x1, .

. ., xc).

Since a monomial is contained in this ideal if and only if it isdivisible by one of the variables (x1, . .

., xc), we may partition F into subsets Fi consistingonly of monomials divisible by xi. The following lemma concludes the proof:Lemma 1.6.

Let Fi ⊆(xi) ⊂S, i = 1, . .

., c be sets of monomials. If F = ∪cj=1Fjgenerates an ideal of codimension c, then the Fj satisfy the hypothesis of Proposition 1.4.Proof.For any subset U as in condition (c), let IU denote the ideal generated by ∪j∈U Fj.The ideal I generated by F is contained in IU + ({xi}i/∈U).

Since I has codimension c,the Principal Ideal Theorem implies that IU has codimension ≥card U as required.Cautionary Example 1.7.Let Fi and in≺(Fi) be as in Theorem 1.3, and choosecoefficients ri,f such that the linear combinations of initial termsXf∈F1r1,f · in≺(f) ,. .

.. . .

,Xf∈Fcrc,f · in≺(f)form a system of parameters. It is tempting to hope that the fi in (∗), made with the samecoefficients ri,f, would also form a system of parameters.

This is not true: If F1 = {x21−x22}and F2 = {x1x2, x22} then x21, x1x2 + x22 is a regular sequence but x21 −x22, x1x2 + x22 is not.We close Section 1 with two propositions showing that our examples 1.1 and 1.2 areminimal in a certain sense. The proofs are straightforward and will be omitted.Proposition 1.8.

Suppose that F ⊂S generates an ideal of codimension c and that Fcannot be partitioned into subsets F1, . .

., Fc such that for some choice of coefficients ri,fthe polynomials (∗) form a system of parameters. (a) The set F can be replaced by a set of c + 1 linear combinations of the elements of Fhaving the same property, possibly after reducing c.(b) Factoring out m −c general linear forms, the number of variables of S may be takento be c.Thus the critical case concerns sets of c+1 polynomials generating an ideal of codimen-sion c in c variables.

It is most interesting to look at the case of homogeneous polynomials.Example 1.2 is of this kind, with c = 4, but there are no such examples with c ≤3:7

Proposition 1.9. If F = {f1, f2, f3, f4} is a set of homogeneous polynomials in 3 vari-ables, generating an ideal of codimension 3 in S = k[x1, x2, , x3], then there is a partitionof F into 3 subsets Fi such that the polynomials in (∗) form a system of parameters.2.

Systems of parameters modulo an idealWe now turn to the general case of our problem, keeping the notation S := k[x1, . .

., xm]as in the introduction. Let J ⊂S be an ideal, let R = S/J and let F ⊂S be a finitesubset.

Let I be the ideal generated by F, and suppose that the codimension of I moduloJ is c in the sense that c = codim(I + J) −codim(J).We say that f1, . .

., fc ∈I is a maximal system of parameters for I modulo J ifcodim(P + (f1, . .

., fc)) ≥codim(I + J) for every minimal prime P of J. (This notion ismost natural if the ideal J is unmixed.) We wish to choose as sparse as possible a maximalsystem of parameters for I modulo J.The simplest and most common problem calling for systems of parameters is that offinding a Noether normalization for a homogeneous ideal.

If c is the Krull dimension of R,this is the problem of finding elements f1, . .

., fc in R such that R is a finitely generatedmodule over the subring k[f1, . .

., fc] ⊆R. (The elements f1, .

. ., fc are then necessarilyalgebraically independent, so that the subring is isomorphic to a polynomial ring.

See[Logar 1988] and [Dickenstein, Fitchas, Giusti, and Sessa 1991] for a discussion from acomputer algebra point of view.) We will focus primarily on this case, but first we presenta method for handling the general problem.

The approach differs from that of Section 1in that it chooses one fi at a time, essentially employing overlapping sets Fi.Greedy Algorithm 2.1.Input :A set of generators F for an ideal I of codimension c modulo J.Let F1 be a minimal subset of F such that F1 is not contained in any minimal prime ofJ of maximal dimension. Let f1 be a sufficiently general combination of the elements ofF1 so that the codimension of J + (f1) is larger than that of J.

Let F′ be the result ofdropping any one of the elements of F1 from F. Replace J by J + (f1), replace F by F′,and iterate the process.Output :The sequence f1, . .

., fc.In case the data F, I, J are homogeneous and the output desired is homogeneous,but not all the polynomials of F are of the same degree, the “sufficiently general linearcombination” would have to have coefficients that are polynomials of varying degrees. Thefollowing variation may be an improvement:Given a minimal subset F1 ⊆F not contained in any minimal prime of J of maximaldimension, replace it with a set of elements whose initial forms are not contained in anyminimal prime of in(J) of maximal dimension.

(This may be done by moving step by step8

toward a Gr¨obner basis of J +(F1), using the Buchberger algorithm, until the codimensionof the initial ideal is larger than that of the initial ideal of J.) Then, if homogeneous outputis desired, each element of F1 not of maximal degree may be multiplied by variables dividingits initial term to bring all the elements of F1 to the same degree before forming the linearcombination as above.We now turn to the special case of Noether normalization.

The following well-knownversion of Hilbert’s Nullstellensatz makes clear the nature of our task:Proposition 2.2. Let J be a homogeneous ideal of S, and set R = S/J.Let X ⊂Pm−1 be the corresponding projective algebraic set.

Suppose that the ground field k isalgebraically closed. If f1, .

. ., fc ∈R are homogeneous polynomials, then R is a finitelygenerated module over the subring k[f1, .

. ., fc] ⊆R if and only if the system of equationsf1(x) = .

. .

= fc(x) = 0 has no solution in X.Proof Sketch:By the Nullstellensatz the condition that there are no solutions is equivalentto the condition that R/(f1, . .

., fc) is a finite dimensional vector space. Because R isgraded and zero in negative degree, a basis for this space lifts to a finite set of generatorsfor R over the subring k[f1, .

. ., fc].In the Noether normalization problem one usually wants the fi to be linear forms.We will henceforth consider only this case, and suppose that F = {x1, .

. .

, xm}, so thatI = (F) is the irrelevant ideal. In this situation Algorithm 2.1 has the effect of reducing ateach step the number of variables to be considered, and this increases its efficiency.

Hereis a monomial example, which also suggests a possibility for improving the algorithm:Example 2.3.Let m = 6, c = 2, F = {x1, x2, x3, x4, x5, x6}, andJ=x1x2, x1x3, x2x3, x2x4, x2x5, x3x4, x3x5, x4x5, x4x6, x5x6)=(x1, x2,x4, x5) ∩(x1, x3, x4, x5) ∩(x2, x3, x4, x5) ∩(x2, x3, x4, x6) ∩(x2, x3, x5, x6).In the first step the unique optimal choice is F1 = {x1, x6}. We set f1 = x1 −x6 andremove x1 from F. Repeating the procedure withJ + (f1)=x1 −x6, x2x3, x2x4, x2x5, x2x6, x3x4, x3x5, x3x6, x4x5, x4x6, x5x6,we must use all remaining variables: F2 = {x2, x3, x4, x5, x6}.

For the second parameterwe can choose, for instance, f2 = x2 + x3 + x4 + x5 + x6.In general Algorithm 2.1 is “too greedy”: it does not perform optimally with respectto sparseness. For example,ˆf1 = x1 + x2 + x3,ˆf2 = x4 + x5 + x69

is also a Noether normalization for the ideal J in Example 2.3. It has a total of only sixnon-zero terms, while the output f1, f2 of Algorithm 2.1 has seven non-zero terms.This shows that the subtlety of sparse Noether normalization is not completely cap-tured by Algorithm 2.1.The remainder of this paper is devoted to a more thoroughcombinatorial analysis, leading to an optimal result.Sparsity of Linear SubspacesWe begin with some remarks on the notion of sparseness itself.

From Proposition 2.2 wesee that the problem of choosing a space of linear forms f1, . .

., fc ∈I of given degree d thatare a homogeneous system of parameters modulo a homogeneous ideal J, is equivalent tothe following geometric problem: given an algebraic set X of dimension c−1 in a projectivespace Pm−1, find a linear subspace L of codimension c not meeting X. Equivalently, wemay think of L as coming from a linear subspace M of an affine space Am, which issupposed to meet the cone over X only in the origin.We wish to choose M to be as sparse as possible, relative to some given system of co-ordinates for Pm−1. There are several possible definitions of sparseness, and they conflictwith one another.

In general, if we agree on a way to represent the space M, then we canspeak of a space allowing the sparsest possible representation in this form. Perhaps thethree most obvious representations are these: M might be represented by the coordinatesof a basis of M (basis representation), by the coordinates of a basis for the space M ⊥oflinear functionals vanishing on M (cobasis representation), or by Pl¨ucker coordinates, themaximal minors of some matrix representing the basis or dual basis (Pl¨ucker representa-tion).

In each case the number of nonzero coordinates is a measure of sparseness — wecall them basis sparseness, cobasis sparseness, and Pl¨ucker sparseness respectively.It is not hard to show that for 1-dimensional subspaces (and thus also for hyperplanes)the three measures of sparseness agree in the sense that all three choose the same spaceas the sparsest in a particular set of subspaces. But in general no two of these measuresagree on naming the sparsest subspace, as may be seen from the following examples.

Ineach case the subspace considered is the row space of the given matrix. As we will notmake use of these facts, we leave their verification to the interested reader.First, to show that Pl¨ucker sparseness does not agree with basis sparseness: The spaceM1 represented by the matrixM1 ↔10000000111111has basis sparseness 7 and Pl¨ucker sparseness 6, while the space M2 represented byM2 ↔1110000000111010

has basis sparseness 6 and Pl¨ucker sparseness 9.The sparseness of a space M is the same as the cobasis sparseness of M ⊥, so thespaces M ⊥1 and M ⊥2 illustrate the same point for cobasis sparseness.It is harder to give examples in which basis sparseness and cobasis sparseness disagree,but the reader may check that if L1 and L2 are the 3-dimensional subspaces of a 9-dimensional vectorspace V represented by the matricesL1 ↔111111111000111222012012101112andL2 ↔111111111000111234012012101112then the basis sparseness of L1 is 6 + 6 + 7 = 19, whereas that of L2 is 6 + 6 + 6 = 18. Thecobasis sparseness of each is 3 + 3 + 3 + 4 + 4 + 4 = 21.

Now consider the subspacesM3 = L1 ⊕L⊥2andM4 = L2 ⊕L⊥1inV ⊕V ∗.The basis sparsenesses of these spaces are 40 = 19 + 21 and 39 = 18 + 21 respectively. Butas M ⊥3 = (L1 ⊕L⊥2 )⊥= L⊥1 ⊕L2, and similarly for M4, the cobasis sparsenesses for M3and M4 are 39 = 18 + 21 and 40 = 19 + 21, reversing the order of the basis sparsenesses.Sparse Noether Normalization using Chow FormsWe now return to the problem of Noether Normalization.

We will work in terms of basissparseness of the space generated by the linear forms in the solution to the Normalizationproblem; our discussion can be adapted, by considering different expressions of the Chowform, to cobasis or Pl¨ucker sparseness as well. Let J be a homogeneous unmixed idealin S, and let X denote its projective variety in Pm−1.

Changing notation somewhat, wesuppose that X has degree p and dimension d−1. We will show how to compute a Noethernormalization consisting of linear formsfi := ci1x1 + ci2x2 + .

. .

+ cimxm ,i = 1, 2, . .

., d(2.1)which is optimal in the sense that the number of non-zero coefficients cij is minimal —that is, the basis sparseness of the space spanned by the fi is minimal. We call the numberof nonzero cij the Noether complexity of X.Let RX = RX(cij) = RX(f1, .

. ., fd) denote the Chow form of X.This classicalpolynomial is characterized by the property that it vanishes if and only if the linear sub-space defined by f1(x) = .

. .

= fd(x) = 0 meets X; see e.g. [Shafarevich 1977, §I.I.6.1]11

and the references given in [Caniglia 1990]. The notation concerning Chow forms tendsto vary from author to author.

The specific notation to be employed here is taken from[Kapranov-Sturmfels-Zelevinsky 1992] and [Sturmfels 1993], namely, we express RX as apolynomial in brackets [ i1 i2 . .

. id ], 1 ≤i1 < .

. .

< id ≤m. These are the d × d-minorsof the d × m-matrix (cij), or, equivalently, the Pl¨ucker coordinates of the codimensionk flat defined by (2.1).

By Proposition 2.2, the Noether normalization problem for R isequivalent to the problem of finding a non-root of the Chow form RX.Example 2.4. (Hypersurfaces, d = m −1)Suppose J is the principal ideal generated by a homogeneous polynomial F(x1, x2, .

. ., xm),defining a hypersurface X ⊂Pm−1.

In terms of brackets its Chow form equalsRX=F[234 . .

.m] , −[134 . .

.m] , [124 . .

.m] , . .

. , (−1)m−1[23 .

. .m −1].

(2.2)The Noether normalizations of X are precisely the (m −1) × m-matrices (cij) for whichthe bracket polynomial (2.2) does not vanish.It is our objective to compute a d × m-matrix (cij), which is a non-root of the Chowform RX, and which is as sparse as possible with this property. Since (cij) must havemaximal rank d, the Noether complexity of X is at least d = dim(X) + 1.

It is exactly dif and only if X is in Noether normal position with respect to some coordinate flat.Observation 2.5. The coordinate forms xi1, xi2, .

. ., xid are a Noether normalization forX if and only if the bracket power [ i1 i2 .

. .

id ]p appears with non-zero coefficient in RX.Let V = {cij : 1 ≤i ≤d, 1 ≤j ≤m} denote the set of variables. For any polynomialf ∈k[V ] we define a simplicial complex ∆(f) as follows: A subset W ⊂V is a face of∆(f) if and only if there exists a non-root of f whose zero coordinates are precisely W.Equivalently, W is not a face of ∆(f) whenever f lies in the ideal generated by W. If wewrite supp(m) for the set of variables dividing a monomial m, we see that the maximalfaces of ∆(f) are the complements of the minimal sets of the form supp(m) where m isa monomial of f. In particular, for each monomial m, the complex ∆(m) is a simplex,consisting of all subsets of V \ supp(m).

Thus we get the first statement of the following:Lemma 2.6. Let f be a homogeneous polynomial in k[V ].

Then ∆(f) is the union ofthe simplices ∆(m), where m ranges over all monomials of f with minimal support. Also,∆(f) is the union of the simplices ∆(in≺(f)), where ≺ranges over all term orders on k[V ].Proof:We have already proved the first statement.

To prove the second it suffices toobserve that because f is homogeneous, every monomial of f with minimal support is theinitial monomial of f with respect to some term order.12

This lemma together with the above observations implies the following theorem.Theorem 2.7. The Noether complexity of a projective variety X equals the least numberof variables cij appearing in any initial monomial in≺(RX) = Q cνijijof its Chow form.Example 2.3.

(continued) The reducible curve X ⊂P5 defined by J has Chow formRX=[ 1 4 ] · [ 1 5 ] · [ 1 6 ] · [ 2 6 ] · [ 3 6 ]=(c11c24 −c14c21) · (c11c25 −c15c21) · (c11c26 −c16c21) · (c12c26 −c16c22) · (c13c26 −c16c23).The coefficient matrices of f1, f2 and ˆf1, ˆf2 considered above are seen to be non-roots ofRX. The Noether complexity of the curve X is equal to six (cf.

Theorem 2.7).The most systematic approach to solving our problem would be to explicitly computethe Chow form RX. By the results of [Caniglia 1990], this can be done in single exponentialtime (in m).

Theorem 2.7 implies that the Noether complexity and an optimal Noethernormalization for X can be computed in single exponential time.Unfortunately, this approach is not useful in practice, since the complete expansion ofthe Chow form into monomials Q cνijijis usually too big. Hence the Caniglia algorithm isonly of theoretical interest with regard to our problem.

In fact, the problem of computingthe Noether complexity of a monomial ideal is NP-hard. The following proof of this facthas been pointed out to us by Jesus DeLoera.

Let G be any graph on V = {x1, . .

., xn}and IG the ideal generated by all square-free cubic monomials xixjxk, and all xixj notcorresponding to an edge of G (this is the Stanley-Reisner ideal of G viewed as a simplicialcomplex). The Noether complexity of IG equals the minimal number 2n −|S1| −|S2|,where S1, S2 ⊂V ranges over all disjoint pairs of stable sets of G. Here Si indexes the zeroentries in row i of a sparsest Noether normalization (cij).

If we had a polynomial timealgorithm for finding S1 and S2, then we could solve the NP-hard problem of computinga maximal stable set in any graph G1 as follows. Let G2 be a disjoint copy of G1, and letG be the graph obtained from their union union G1 ∪G2 by connecting each vertex of G1with each vertex of G2.

Applying our algorithm to G we obtain a maximal stable set Sifor G = Gi, i = 1, 2.For practical computations we propose an approach using (truncated) Gr¨obner basiscomputations for the ideal J. The following proposition, which is easily derived from theproof of Lemma 2.6, shows that the Noether complexity of a homogeneous ideal is boundedabove by the Noether complexity of any of its initial ideals.Proposition 2.8.

Let (cij) be any Noether normalization of the initial monomial idealinω(J), where ω = (ω1, . .

., ωn) ∈Zm represents any term order for J. Then, for almostall t ∈k, the matrix (cij · tωj) is a Noether normalization of J.From this we get the following lifting algorithm: Choose any term order on S, andcompute a truncated Gr¨obner basis {g1, .

. .

, gr} for J, subject to the truncation condition13

that the monomial ideal L =in(f1), . .

., in(fr)has the same radical as inω(J). Computethe Chow form RL of L, e.g., using method in Proposition 3.4 of [Sturmfels 1992].

Thebracket monomial RL has precisely the same bracket factors as Rin(J). We have∆(RL)=∆(Rin(J))⊆∆(RX).

(2.3)Choose any maximal face of the simplicial complex ∆(RM). This gives rise to a Noethernormalization for in(J) and, using Proposition 2.8, we get a Noether normalization for J.In order to find a sparser Noether normalization we may repeat this procedure for asmany different term orders as we can.

In fact, whenever this is feasible, one might like tocompute a universal Gr¨obner basis U for J, that is, a finite subset of J which is a Gr¨obnerbasis simultaneously for all term orders on S. From a universal Gr¨obner basis and theknowledge of the maximal faces of ∆(RM) we can read offthe minimum of the Noethercomplexities of all initial ideals of J. However, this minimum will generally not agree withthe Noether complexity of J, as the following example shows.Cautionary Example 2.9.

A homogeneous ideal J whose Noether complexity is smallerthan the Noether complexity of any of its initial ideals in(J). Let m = 6 and considerJ=(x2x5 −x1x6, x3, x4) ∩(x1x4 −x3x5, x2, x6) ∩(x3x6 −x2x4, x1, x5).The variety X of J is a union of three toric surfaces in P5.

Here the Chow form equalsRX=[126][156] −[125][256]·[135][345] −[145][134]·[234][246] −[236][346].The matrix(cij)=100100010010001001is a non-root of RX and hence defines a Noether normalization. It is optimal because eachterm appearing in the complete expansion of RX contains at least six of the variables cij.This proves that the Noether complexity of X equals six.The ideal J has six distinct initial ideals, each of which is isomorphic toin(J)=(x2x5, x3, x4) ∩(x1x4, x2, x6) ∩(x2x4, x1, x5).The complete expansion of the initial Chow formRin(J)=[126][156][135][345][236][346]has 13, 452 terms.

Each of these terms contains at least eight variables. Hence the Noethercomplexity of each initial ideal of J equals eight.14

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