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From Oil Fields to Hilbert Schemes

Lorenzo Robbiano

Università di Genova Dipartimento di Matematica

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 1 / 31

Two styles of presentation MATHEMATICIAN

• M. Kreuzer and L. Robbiano, Deformations of border bases,

arXiv:0710.2641. To appear on “Collectanea Mathematica".

• L. Robbiano, On border basis and Gröbner basis schemes,

arXiv:0802.2793. To appear on “Collectanea Mathematica". NOVELIST The job of drilling wells inside oil fields inspired his mathematical aptitude...

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 2 / 31

Facts

In the realm of polynomial algebra two main ingredients need manipulation and implementation, discrete and continuous data. Every polynomial over R or C is built on top of a discrete object, the support, and a continuous object, the set of its coefficients. The support is well understood. If the coefficients are not exact, the very notion of a polynomial, and all the derived algebraic structures (ideals, free resolutions, Hilbert functions,...), tend to be blurred.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 3 / 31

Facts II

Easy example: consider three non-aligned points in the affine plane over the reals. Meaning of being non-aligned? A better description should be being far from aligned? The vanishing ideal is generated by three quadratic polynomials. If we change some of the coefficients of these polynomials by a small amount, almost surely we get the unit ideal. A new fields of investigation is emerging. We have named it Approximate Commutative Algebra (ApCoA) http://cocoa.dima.unige.it/conference/apcoa2008/

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 4 / 31

Facts III

Approximate coefficients may encode experimental data like measures of physical quantities inside an oil field.

http://staff.fim.uni-passau.de/algebraic-oil/en/index.html

If we want to use algebraic methods to build polynomial models, we face the difficulty of doing good multivariate interpolation. Gröbner bases are not well-suited because of the rigid structure imposed by term orderings. Other objects behave better, are called border bases. There is a link between border bases and Hilbert schemes.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 5 / 31

Contents

1

Interpolation on Finite Sets of Points

2

Data Affected by Errors

3

Border Bases

4

Families of Border Bases

5

Gröbner and Border Basis Schemes References

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 6 / 31

Interpolation on Finite Sets of Points

Separators and Interpolators I

We start with a finite set of points X in the affine space and call it an affine point set. Its vanishing ideal in P is called I(X). If we want to perform polynomial interpolation, we also need to know the following polynomials. Definition Let X = {p1, . . . , ps} ⊆ K n be an affine point set, and let X be the tuple (p1, . . . , ps). Let i ∈ {1, . . . , s}. A polynomial f ∈ P is called a separator of pi from X \ pi if f(pi) = 1 and f(pj) = 0 for j = i . Let a1, . . . , as ∈ K . A polynomial f ∈ P is called an interpolator for the tuple (a1, . . . , as) at X if f(pi) = ai for i = 1, . . . , s.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 8 / 31

Interpolation on Finite Sets of Points

Separators and Interpolators II

Proposition Let X = {p1, . . . , ps} ⊆ K n be an affine point set, and let X be the tuple (p1, . . . , ps). For every i ∈ {1, . . . , s}, there exists a separator of pi from X \ pi . For every (a1, . . . , as) ∈ K s , there exists an interpolator for (a1, . . . , as) at X .

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 9 / 31

Interpolation on Finite Sets of Points

The Buchberger-Möller Algorithm

Theorem (The Buchberger-Möller Algorithm) Let σ be a term ordering on Tn , and let X = {p1, . . . , ps} be an affine point set in K n whose points pi = (ci1, . . . , cin) are given via their coordinates cij ∈ K . Consider the following sequence of instructions.

1) Let G = ∅ , O = ∅ , S = ∅ , L = {1} , and let M = (mij ) ∈ Mat0,s(K) be a matrix having s columns and initially zero rows. 2) If L = ∅ , return the pair (G, O) and stop. Otherwise, choose the term t = minσ(L) and delete it from L . 3) Compute the evaluation vector (t(p1), . . . , t(ps)) ∈ Ks and reduce it against the rows of M to obtain (v1, . . . , vs) = (t(p1), . . . , t(ps)) − P i ai (mi1, . . . , mis) with ai ∈ K 4) If (v1, . . . , vs) = (0, . . . , 0) then append the polynomial t − P i ai si to G where si is the ith element in S . Remove from L all multiples of t . Then continue with step 2). 5) Otherwise (v1, . . . , vs) = (0, . . . , 0) , so append (v1, . . . , vs) as a new row to M and t − P i ai si as a new element to S . Add t to O , and add to L those elements of {x1t, . . . , xnt} which are neither multiples of an element of L nor of LTσ(G) . Continue with step 2).

This is an algorithm which returns (G, O) such that G is the reduced σ-Gröbner basis of I(X) and O = Tn \ LTσ{I(X)}. A small alteration of this algorithm allows us to compute the separators of X as well.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 10 / 31

Interpolation on Finite Sets of Points

Some Remarks

Using Buchberger-Möller algorithm it is possible to compute ideals of points and to solve the problem of interpolation. Separators and interpolators are not unique. Two separators of pi and two interpolators for a tuple (a1, . . . , as) ∈ K s differ by an element of I(X).

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 11 / 31

Data Affected by Errors

Bases of P/I

Approximate versions of the above results should be able to construct sets of polynomials which almost vanish at X, almost separators and almost interpolators. New problems of stability arise in this context. The eigenvalue method uses multiplication matrices which require the choice of a basis of A = P/I as a K -vector space. If σ is a term ordering on Tn and G = {f1, . . . , fs} is a σ-Gröbner basis

• f I , then LTσ{I} = {LTσ(f1), . . . , LTσ(fs)}. We know that the residue

classes of the elements of Tn \ LTσ{I} form a K -basis of A. If we change σ we may get different bases of A. Question 1 Are these the only bases? Question 2 What are the most stable bases?

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 13 / 31

Data Affected by Errors

Two conics I

Example Consider the polynomial system f1 =

1 4 x2 + y2 − 1

= f2 = x2 + 1

4 y2 − 1

= X = Z(f1) ∩ Z(f2) consists of the four points X = {(±

• 4/5, ±
• 4/5)}.

The set {x2 − 4

5, y2 − 4 5} is the reduced Gröbner basis of the ideal

I = (f1, f2) ⊆ C[x, y] with respect to σ = DegRevLex. Therefore we have LTσ(I) = (x2, y2), and the residue classes of the terms in T2 \ LTσ{I} = {1, x, y, xy} form a C-vector space basis of C[x, y]/I .

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 14 / 31

Data Affected by Errors

Two conics II

Now consider the slightly perturbed polynomial system ˜ f1 =

1 4 x2 + y2 + ε xy − 1

= ˜ f2 = x2 + 1

4 y2 + ε xy − 1

= where ε is a small number. The intersection of Z(˜ f1) and Z(˜ f2) consists of four perturbed points X close to those in X.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 15 / 31

Data Affected by Errors

Two conics III

This time the ideal ˜ I = (˜ f1,˜ f2) has the reduced σ-Gröbner basis {x2 − y2, xy + 5

4ε y2 − 1 ε, y3 − 16ε 16ε2−25 x + 20 16ε2−25 y}

Moreover, we have LTσ(˜ I) = (x2, xy, y3) and T2 \ LTσ{˜ I} = {1, x, y, y2}. A small change in the coefficients of f1 and f2 has led to a big change in the Gröbner basis of (f1, f2) and in the associated vector space basis

• f C[x, y]/(f1, f2), although the zeros of the system have not changed

much. Numerical analysts call this kind of unstable behaviour a representation singularity.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 16 / 31

Border Bases

Border Bases: Introduction

The basic idea of border basis theory is to describe a zero-dimensional ring P/I by an order ideal of monomials O whose residue classes form a K -basis

• f P/I and by the multiplication tables of this basis.

Let K be a field, let P = K[x1, . . . , xn], and let Tn be the monoid of terms. Definition A set of polynomials G = {g1, . . . , gν} in P is called an O-border prebasis if the polynomials have the form gj = bj − µ

i=1 αijti with αij ∈ K for 1 ≤ i ≤ µ,

1 ≤ j ≤ ν, bj ∈ ∂O, ti ∈ O. Definition Let G = {g1, . . . , gν} be an O-border prebasis, and let I ⊆ P be an ideal containing G. The set G is called an O-border basis of I if the residue classes O = {¯ t1, . . . ,¯ tµ} form a K -vector space basis of P/I .

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 18 / 31

Border Bases

Existence and uniqueness

Proposition (Existence and Uniqueness of Border Bases) Let O = {t1, . . . , tµ} be an order ideal, let I ⊆ P be a zero-dimensional ideal, and assume that the residue classes of the elements of O form a K -vector space basis of P/I . Then there exists a unique O-border basis of I . Proposition (Border Bases and Gröbner Bases) Let σ be a term ordering on Tn , and let Oσ(I) be the order ideal Tn \ LTσ{I}. Then there exists a unique Oσ(I)-border basis G of I , and the reduced σ-Gröbner basis of I is the subset of G corresponding to the corners

• f Oσ(I).

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 19 / 31

Border Bases

Multiplication matrices

Border bases can be characterized imitating Gröbner basis theory. So we can use special generation, rewrite relations, syzygies and an important new feature, described by Mourrain. Theorem (Border Bases and Commuting Matrices) Let O = {t1, . . . , tµ} be an order ideal, let G = {g1, . . . , gν} be an O-border prebasis, and let I = (g1, . . . , gν). Then the following conditions are equivalent.

1

The set G is an O-border basis of I .

2

The formal multiplication matrices of G are pairwise commuting.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 20 / 31

Border Bases

Two conics IV

What are the border bases in the two cases of the conics and the perturbed conics? Two conics {x2 − 4

5,

x2y − 4

5y,

xy2 − 4

5x,

y2 − 4

5}

Two perturbed conics {x2 + 4

5 εxy − 4 5,

x2y −

16ε 16ε2−25 x + 20 16ε2−25 y,

xy2 +

20 16ε2−25 x + 16ε 16ε2−25 y,

y2 + 4

5 εxy − 4 5}

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 21 / 31

Families of Border Bases

BB Schemes

The following material is taken from

• M. Kreuzer, L. Robbiano, Deformations of border bases,

arXiv:math\0710.2641v1. To appear on “Collectanea Mathematica". Let O = {t1, . . . , tµ} be an order ideal in Tn , and let ∂O = {b1, . . . , bν} be its border. We construct a universal family for all zero-dimensional ideals having an O-border basis. Definition Let {cij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν} be a set of further indeterminates.

1

The generic O-border prebasis is the set of polynomials G = {g1, . . . , gν} in Q = K[x1, . . . , xn, c11, . . . , cµν] given by gj = bj − µ

i=1 cijti

2

For k = 1, . . . , n, let Ak ∈ Matµ(K[cij]) be the kth formal multiplication matrix associated to G. Then the affine scheme BO ⊆ K µν defined by the ideal I(BO) generated by the entries of the matrices AkAℓ − AℓAk with 1 ≤ k < ℓ ≤ n is called the O-border basis scheme. The ideal I(BO) is generated by quadratic polynomials.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 23 / 31

Families of Border Bases

Philosophy

A border basis of an ideal of points I in P is intrinsically related to a basis O of the quotient ring. If we move the points slightly, O is still a basis of the perturbed ideal ˜ I , since the evaluation matrix of the elements of O at the points has determinant different from zero. Moving the points moves the border basis, and the movement traces a path inside the border basis scheme. On the other hand, if we perturb the equations of the border basis, in general the multiplication matrices almost commute, but most likely the new ideal is the unit ideal. The philosophical question here is the following. When we compute with approximate data, do we want to move along the border scheme, or are we content to simply stay close to the border scheme?

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 24 / 31

Families of Border Bases

Border Bases and Hilbert Schemes

Here we collect some basic observations about border basis schemes. BO can be embedded as an open affine subscheme of the Hilbert scheme which parametrizes subschemes of An of length µ (see [MS]). There is an irreducible component of BO of dimension n µ which is the closure of the set of radical ideals having an O-border basis. The dimension of BO is claimed to be nµ in [St2]. But Iarrobino’s example which features a high-dimensional component of the Hilbert scheme, yields a counterexample to this claim. It follows that the border basis scheme is in general reducible. In the case n = 2 more precise information is available: for instance, it is known that BO is reduced, irreducible and smooth of dimension 2µ (see [Ha], [Hu] and [MS]). For every term ordering σ, there is a subset of BO which parametrized all ideals I such that O = Oσ(I). These subsets have turned out to be useful for studying the Hilbert scheme which parametrizes subschemes

• f An of length µ (see for instance [CV] and [NS]).

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 25 / 31

Families of Border Bases

Connectedness

Can we deform ideals in BO to the unique monomial ideal in BO ? The direct approach is to imitate Gröbner basis theory and to use the flat deformation to the degree form ideal. This approach does not succeed in all cases, but ... it does, under the additional assumption that O has a maxdegW border, i.e. that no term in O has a larger degree than a term in the border ∂O.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 26 / 31

Gröbner and Border basis Schemes

The Gröbner Scheme and the Universal Family

The following material is taken from

• L. Robbiano, On Border Basis and Gröbner Basis Schemes,

arXiv:0802.2793. The main idea is to compare Gröbner basis schemes with border basis schemes. First, I define the Gröbner basis scheme GO,σ in a way which is different from the usual ones. Then I also define a universal family over the Gröbner basis scheme. The first main result says that Gröbner basis schemes and their associated universal families can be endowed with a graded structure where the indeterminates have positive weights. In other words, they can be viewed as weighted projective schemes. The second main result is a theorem where the comparison of the two schemes is fully described. In particular, it is shown that Gröbner basis schemes can be obtained as sections of border basis schemes with suitable linear spaces.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 28 / 31

Gröbner and Border basis Schemes

Example Let O = {1, x, y, xy}. We observe that t1 = 1, t2 = x , t3 = y , t4 = xy , b1 = x2, b2 = y2 , b3 = x2y , b4 = xy2 . Let σ = DegRevLex, so that x >σ y . g1 = x2 − c111 − c21x − c31y − c41xy g2 = y2 − c121 − c22x − c32y − c42xy g3 = x2y − c131 − c23x − c33y − c43xy g4 = xy2 − c141 − c24x − c34y − c44xy Then necessarily c42 = 0 (the linear space), so that g2 is replaced by g∗

2 = y2 − c121 − c22x − c32y

To make everything compatible with σ we need x > y , y2 > x . We put deg(x) = 3, deg(y) = 2 To make everything homogeneous we put deg(c11) = 6, deg(c21) = 3, deg(c31) = 4, deg(c41) = 1, . . .

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 29 / 31

Gröbner and Border basis Schemes

The Gröbner Scheme II

Corollary Let O ⊂ Tn be an order ideal of monomials, let σ be a term ordering on Tn , and let o be the origin in the affine space As(cO,σ) . The point o belongs to GO,σ . The following conditions are equivalent

1) The scheme GO,σ is isomorphic to an affine space. 2) The point o is a smooth point of GO,σ .

In the literature Gröbner basis schemes are mostly described using Buchberger’s Algorithm. However, the reduction process in the algorithm is far from being unique, hence the description of the Gröbner basis scheme is a priori not canonical. With our approach we prove that all the ideals obtained using Buchberger’s Algorithm are equal and coincide with the ideal defined in the paper.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 30 / 31

Gröbner and Border basis Schemes

Two open problems

We know that GO,σ can be obtained as a linear section of BO . Question 1: Is there any connection between the smoothness of the origin in GO,σ and the smoothness of the origin in BO ? The scheme GO,σ is connected since it is a quasi-cone, and hence all its points are connected to the origin. We know the precise relation between the two schemes GO,σ and BO . However, the problem of the connectedness of BO is still open. Question 2: Is BO connected?

Judge others by their questions rather than by their answers. (François-Marie Arouet (Voltaire))

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 31 / 31

Gröbner and Border basis Schemes

References I

• J. Abbott, A. Bigatti, M. Kreuzer, L. Robbiano Computing Ideals of

Points J. Symb. Comput. Vol 30, pp 341–356 (2000).

• J. Abbott, C. Fassino, M. Torrente, Thinning out redundant empirical data

Mathematics in Computer Science, vol. 1 no. 2, pp. 375–392 (2007).

• J. Abbott, C. Fassino and M. Torrente, Stable border bases for ideals of

points To appear on J. Symb. Comput..

• J. Abbott, M. Kreuzer, L. Robbiano Computing zero-dimensional

Schemes J. Symb. Comput. Vol 39, pp 31–49 (2005).

• B. Buchberger, Groebner Bases: An Algorithmic Method in Polynomial

Ideal Theory, in: (N.K. Bose, Ed.) Multidimensional Systems Theory. D. Reidel Publ. Comp. Pp. (1985) 184–232

• M. Caboara and L. Robbiano Families of Estimable Terms Proc. ISSAC

’01, 56–63 (2001)

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 32 / 31

Gröbner and Border basis Schemes

References II

• G. Carrà and L. Robbiano On superG-bases,
• J. Pure Appl. Algebra 68, 279–292 (1990)
• A. Conca and G. Valla, Canonical Hilbert-Burch matrices for ideals of

k[x, y], arXiv:math\0708.3576. The CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.

• D. Cox, Solving equations via algebras, in: A. Dickenstein and I. Emiris

(eds.), Solving Polynomial Equations, Springer, Berlin (2005). T.S. Gustavsen, D. Laksov and R.M. Skjelnes, An elementary, explicit, proof of the existence of Hilbert schemes of points, arXiv:math\0506.161v1.

• M. Haiman, q,t-Catalan numbers and the Hilbert scheme, Discr. Math.

193 (1998), 201–224.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 33 / 31

Gröbner and Border basis Schemes

References III

• D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse, Approximate

Computation of Zero-Dimensional Polynomial Ideals, Preprint (2006).

• M. Huibregtse, A description of certain affine open schemes that form an
• pen covering of Hilbn

A2

k , Pacific J. Math. 204 (2002), 97–143.

• M. Huibregtse, An elementary construction of the multigraded Hilbert

scheme of points, Pacific J. Math. 223 (2006), 269–315.

• A. Kehrein and M. Kreuzer, Characterizations of border bases, J. Pure
• Appl. Alg. 196 (2005), 251–270.
• A. Kehrein and M. Kreuzer, Computing border bases, J. Pure Appl. Alg.

205 (2006), 279–295.

• A. Kehrein, M. Kreuzer and L. Robbiano, An algebraist’s view on border

bases, in: A. Dickenstein and I. Emiris (eds.), Solving Polynomial Equations: Foundations, Algorithms, and Applications, Springer, Heidelberg 2005, 169–202.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 34 / 31

Gröbner and Border basis Schemes

References IV

• M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1,

Springer, Heidelberg (2000).

• M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2,

Springer, Heidelberg (2005).

• M. Kreuzer and L. Robbiano, Deformations of border bases,

arXiv:0710.2641. To appear on “Collectanea Mathematica".

• A. Iarrobino, Reducibility of the families of 0-dimensional schemes on a

variety, Invent. Math. 15 (1972), 72–77.

• B. Mourrain, A new criterion for normal form algorithms, AAECC Lecture

Notes in Computer Science 1719 (1999), 430–443.

• E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Springer,

New York 2005.

• R. Notari and M. L. Spreafico, A stratification of Hilbert schemes by initial

ideals and applications, Manuscripta Math. 101 (2000), 429–448.

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 35 / 31

Gröbner and Border basis Schemes

References V

• L. Robbiano, On border basis and Gröbner basis schemes,

arXiv:0802.2793. To appear on “Collectanea Mathematica".

• H. Stetter, “Approximate Commutative Algebra" - an ill-chosen name for

an important discipline, ACM Communications in Computer Algebra, Vol 40, N0 3 (2006).

• H. Stetter, Numerical Polynomial Algebra, SIAM, Philadelphia (2004).

Lorenzo Robbiano (Università di Genova) From Oil Fields to Hilbert Schemes June, 2008 36 / 31