Polynomial Models: from data to equations, from equations to solutions

Polynomial Models: from data to equations, from equations to solutions Lorenzo Robbiano Università di Genova Dipartimento di Matematica

Lorenzo Robbiano (Università di Genova)

Polynomial Models

Torino Politecnico, 2009

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Contents

Introduction 1

The Algebraic Setting: Polynomials and Gröbner Bases

2

From Equations to Points: the Lex Method

3

From Equations to Points: Eigenvalues and Eigenvectors

4

From Points to Equations: the Buchberger-Möller Algorithm

5

Border Bases

6

The Real World: Approximation

7

Families of Border Bases

References


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Introduction In this mini-course we are going to discuss the basic techniques for solving polynomial systems. Then we do some work on interpolation and introduce border bases. Finally we see some connections between border bases, approximation, and possibly Hilbert schemes. Examples are shown with the help of CoCoA. [CoCoA-Web] [open CoCoA] Standard references: M. Kreuzer – L. Robbiano: Computational Commutative Algebra 1, Springer (2000) [Book1] M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005) [Book2]


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A Multizone Well


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Soft Spots


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Polynomial systems Suppose we are given f1 , . . . , fs in the ring P = K [x1 , . . . , xn ] over a field K . We want to solve the following system of polynomial equations:   f1 (x1 , . . . , xn ) = 0 .. .   fs (x1 , . . . , xn ) = 0 If the polynomials f1 , . . . , fs have degrees = 1 , how to solve it is a well-known result in Linear Algebra. Moreover, the size of the set of solutions does not depend on the field K . However, as soon as at least one of the polynomials f1 , . . . , fs has degree ≥ 2 , it turns to be more appropriate to look for the set of n solutions in K , where K is the algebraic closure of K . A first observation is that solving the above system of equations means to determine the set of zeros of the ideal I = (f1 , . . . , fs ) . Lorenzo Robbiano (Università di Genova)

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Notation In the sequel, let K be a field, let P = K [x1 , . . . , xn ] , let f1 , . . . , fs ∈ P , and let I = (f1 , . . . , fs ) . Moreover, let K be the algebraic closure of K , and let P = K [x1 , . . . , xn ] . By S we shall denote the system of polynomial equations   f1 (x1 , . . . , xn ) = 0 .. .   fs (x1 , . . . , xn ) = 0 n

We recall that while Z(I) denotes the set of all the zeros of I in K , we denote by ZK (I) the set of all the zeros of I in K n . Given a system S as above, there can be finitely or infinitely many n solutions in K , i.e. the set Z(I) may be finite or infinite. Next proposition provides an algorithmic criterion for finiteness.


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Finiteness Criterion

Proposition (Finiteness Criterion) Let σ be a term ordering on Tn. The following conditions are equivalent. 1

The system of equations S has only finitely many solutions.

2

The ideal I P is contained in only finitely many maximal ideals of P.

3

For i = 1, . . . , n , we have I ∩ K [xi ] 6= (0).

4

The K -vector space K [x1 , . . . , xn ]/I is finite-dimensional.

5

The set Tn \ LTσ {I} is finite.

6

For every i ∈ {1, . . . , n} , there exists a number αi ≥ 0 such that we have xiαi ∈ LTσ (I) .


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Zero-dimensional ideals Definition An ideal I = (f1 , . . . , fs ) in P = K [x1 , . . . , xn ] is called zero-dimensional if it satisfies the equivalent conditions of the Finiteness Criterion. Corollary With the same assumptions and notation as in the Finiteness Criterion, let I and J be ideals in P . 1

If I is maximal, then I is zero-dimensional.

2

If I is zero-dimensional and I ⊆ J , then J is zero-dimensional.

3

If I is zero-dimensional, then IP is also zero-dimensional and dimK (P/I) = dimK (P/IP)

We shall assume from now on that there are only finitely many such solutions, i.e. that the ideal I = (f1 , . . . , fs ) is zero-dimensional. Lorenzo Robbiano (Università di Genova)

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How many solutions? The Finiteness Criterion yields an easy bound for the number of solutions of S . For, we use condition (3) and deduce that the number of solutions of S is at most deg(g1 ) · · · deg(gn ) . In fact, a much sharper bound is available and to prove it we need a ring-theoretic version of the Chinese Remainder Theorem. Proposition (Bound for the Number of Solutions) Let f1 , . . . , fs ∈ P generate a zero-dimensional ideal I = (f1 , . . . , fs ) . Then the system of equations f1 (x1 , . . . , xn ) = · · · = fs (x1 , . . . , xn ) = 0 n

has at most dimK (P/I) solutions in K .


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How many solutions? Example Let us consider the three polynomials f1 = x 2 + y + z − 1 , f2 = x + y 2 + z − 1 , and f3 = x + y + z 2 − 1 in P = Q[x, y , z] . They define a system of polynomial equations f1 = f2 = f3 = 0 and generate an ideal I = (f1 , f2 , f3 ) in P . [Example1] We observe that {f1 , f2 , f3 } is a DegRevLex-Gröbner Basis of I and deduce that LTDegRevlex (I) = (x 2 , y 2 , z 2 ) , so that dimQ (P/I) = 8 . Therefore 8 is an upper bound for the number of solutions. How sharp is this a bound? When we compute generators gi of the elimination ideals I ∩ Q[xi ] for i = 1, 2, 3 and factor them, we get √ √ g1 = x 6 − 4x 4 + 4x 3 − x 2 = x 2 (x − 1)2 (x + 1 + √2)(x + 1 − √2) g2 = y 6 − 4y 4 + 4y 3 − y 2 = y 2 (y − 1)2 (y + 1 + √2)(y + 1 − √ 2) g3 = z 6 − 4z 4 + 4z 3 − z 2 = z 2 (z − 1)2 (z + 1 + 2)(z + 1 − 2) Each of those polynomials has four different zeros. By substituting them into the original system of equations, we see that of the only the √ 64 possible √ combinations √ five tuples √ {(1, 0,√0), (0, 1, 0), √ (0, 0, 1), (−1 + 2, −1 + 2, −1 + 2), (−1 − 2, −1 − 2, −1 − 2)} are actual solutions. Lorenzo Robbiano (Università di Genova)

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Radical ideals In this example, we see other phenomena emerging. For instance, it is clear that the last step of the procedure applied in this example breaks down if the zeros of the polynomials g1 , . . . , gn cannot be represented by radicals. Another important fact is that, while the bound given by the Proposition is 8 , the actual number of solutions is 5 . If a polynomial f vanishes at the set of solutions of S , then also sqfree(f ) vanishes at that set. Consequently, the system S has the same set of solutions as the system S 0 , where S 0 is obtained from S by adding the squarefree parts of some polynomials in S . Looking at the example above, we can now understand why the upper bound given by the Proposition is not sharp. √ √ We have Z(I) = Z( I) , and dimK (P/I) ≥ dimK (P/ I) . √ The radical of I , i.e. I can be computed if K has charachteristic zero or is a perfect field of characteristic p > 0 having effective pth roots. Lorenzo Robbiano (Università di Genova)

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Exact number of solutions

If the system of polynomial equations S corresponds to a zero-dimensional radical ideal, and if the base field is perfect, the bound for the number of solutions given in the Proposition is sharp, i.e. we have the following formula for the exact number of solutions. Theorem (Exact Number of Solutions) Let I be a zero-dimensional radical ideal in P , let K be the algebraic closure of K , and let P = K [x1 , . . . , xn ] . If K is a perfect field, the number of solutions of the system of equations S is equal to the number of maximal ideals of P containing IP , and this number is precisely dimK (P/I) .


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From equations to points: the Lex method

Normal position The Lex method is based on the idea of extending the technique of Gaussian elimination. We want to solve a system f1 = · · · = fs = 0 and may assume that I = (f1 , . . . , fs ) is a zero-dimensional radical ideal in P = K [x1 , . . . , xn ] . Our next goal is to perform a linear change of coordinates in such a way that the resulting system of equations has the additional property that its n solutions in K have pairwise distinct last coordinates. Let us introduce the following name for this property. Definition Suppose that I is a zero-dimensional ideal in the polynomial ring P , and let i ∈ {1, . . . , n} . We say that I is in normal xi -position if any two zeros n (a1 , . . . , an ), (b1 , . . . , bn ) ∈ K of I satisfy ai 6= bi . If the field K has enough elements (in particular if it is infinite), a linear transformation can be explicitly computed which puts the ideal I in normal xi -position. Lorenzo Robbiano (Università di Genova)

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Shape

Theorem (The Shape Lemma) Let K be a perfect field, let I ⊆ P be a zero-dimensional radical ideal in normal xn -position, let gn ∈ K [xn ] be the monic generator of the elimination ideal I ∩ K [xn ] , and let d = deg(gn ) . 1

The reduced Gröbner basis of the ideal I with respect to Lex is of the form {x1 − g1 , . . . , xn−1 − gn−1 , gn } , where g1 , . . . , gn−1 ∈ K [xn ] .

2

The polynomial gn has d distinct zeros a1 , . . . , ad ∈ K , and the set of zeros of I is Z(I) = {(g1 (ai ), . . . , gn−1 (ai ), ai ) | i = 1, . . . , d}


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Solving Corollary (Solving Systems Effectively) Let K be a field of characteristic zero or a perfect field of characteristic p > 0 having effective pth roots. Furthermore, let f1 , . . . , fs ∈ P = K [x1 , . . . , xn ] , and let I = (f1 , . . . , fs ) . Consider the following sequence of instructions. 1

For i = 1, . . . , n , compute a generator gi Infinite Solution Set and stop.

2

Compute hi = sqfree(gi ) for i = 1, . . . , n , then replace I by I + (h1 , . . . , hn ) . Compute d = #(Tn \ LTσ {I}) .

3 4 5

of the elimination ideal I ∩ K [xi ] . If gi = 0 for some i ∈ {1, . . . , n} , then return

Check if deg(hn ) = d . In this case, let (c1 , . . . , cn−1 ) = (0, . . . , 0) and continue with step 8). “ ” d If K is finite, enlarge it so that it has more than elements. 2

6

Choose a tuple (c1 , . . . , cn−1 ) ∈ K n−1 . Apply the coordinate transformation x1 7→ x1 , . . . , xn−1 7→ xn−1 , xn 7→ xn − c1 x1 − · · · − cn−1 xn−1 to I and get an ideal J .

7

Compute a generator of J ∩ K [xn ] and check if it has degree d . If not, repeat steps 6) and 7) until this is the case. Then rename J and call it I .

8

Compute the reduced Gröbner basis of I with respect to Lex. It has the shape {x1 − g1 , . . . , xn−1 − gn−1 , gn } with polynomials g1 , . . . , gn ∈ K [xn ] and with deg(gn ) = d . Return the tuples (c1 , . . . , cn−1 ) and (g1 , . . . , gn ) and stop. This is an algorithm which decides whether the system of polynomial equations S given by f1 = · · · = fs = 0 has finitely many solutions. In that case, it returns tuples (c1 , . . . , cn−1 ) ∈ K n−1 and (g1 , . . . , gn ) ∈ K [xn ]n such that, after we perform the linear change of coordinates x1 7→ x1 , . . . , xn−1 7→ xn−1 , xn 7→ xn − c1 x1 − · · · − cn−1 xn−1 , the transformed system of equations has the set of solutions {(g1 (ai ), . . . , gn−1 (ai ), ai ) | i = 1, . . . , d} , where a1 , . . . , ad ∈ K are the zeros of gn . In other words, the original system of equations has the set of solutions {(g1 (ai ), . . . , gn−1 (ai ), ai − c1 g1 (ai ) − · · · − cn−1 gn−1 (ai )) | i = 1, . . . , d}


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From equations to points: Eigenvalues and Eigenvectors

Multiplication map As before, let K be a field, let P = K [x1 , . . . , xn ] , and let f1 , . . . , fs ∈ P , let K be the algebraic closure of K , and let P = K [x1 , . . . , xn ] . By S we denote the system of polynomial equations   f1 (x1 , . . . , xn ) = 0 .. .   fs (x1 , . . . , xn ) = 0 By I we denote the ideal (f1 , . . . , fs ) , by A the quotient ring P/I , and by A the quotient ring P/IP . As usual we assume that I is zero-dimensional so that A is a finite dimensional K -vector space by the Finiteness Criterion. Definition Given a polynomial f ∈ P we consider the following K -linear map mf : A −→ A defined by mf (g mod I) = fg mod I , and call it the multiplication map defined by f . We also consider the induced K -linear map on the dual spaces mf∗ : A∗ −→ A∗ defined by mf∗ (ϕ) = ϕ ◦ mf . Lorenzo Robbiano (Università di Genova)

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A first example Example Let P = R[x] , f = x 2 + 1 , I the ideal generated by the polynomial x 3 − x 2 + x − 1 = (x − 1)f , and A = P/I . We consider the multiplication map mf : A −→ A and describe it via its representation with respect to the R -basis of A given by (1, x, x 2 ) . Using the relations x 3 + x = x 2 + 1 mod I , x 4 + x 2 = x 2 + 1 mod I , we deduce that the matrix which represents mf is 0

1 @0 1

1 0 1

1 1 0A 1

It is singular, hence 0 is an eigenvalue. Moreover, Z(I) = {1, i, −i} , and we see that 0 = f (i) = f (−i) , but ZR (I) = {1} and 0 6= f (1) . Next theorem makes an important link between Z(I) and eigenvalues, and the example motivates the reason why we need the assumption that ZK (I) = Z(I) . Lorenzo Robbiano (Università di Genova)

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Eigenvalues Theorem Let I be a zero-dimensional ideal in P such that ZK (I) = Z(I) , let f ∈ P , and let λ ∈ K . The following conditions are equivalent 1 2

The element λ is an eigenvalue of mf . There exists a point p ∈ Z(I) such that λ = f (p) .

Proof. Let id : A −→ A denote the identity map. We know that λ is an eigenvalue of mf Let us prove (1) =⇒ (2) .

if and only if mf − λ id is not invertible.

If λ does not coincide with any of the values of f Z(J) = ∅ .

at the points p ∈ Z(I) , then the ideal J = I + (f − λ) has the property that

On the other hand the ideal J is defined over K , hence the Weak Nullstellenzatz implies that 1 ∈ J . It means that there exists g ∈ P such that 1 = g(f − λ) + h with h ∈ I . Then 1 = g(f − λ) mod I , so that mf − λ id is invertible with inverse mg , and hence λ is not an eigenvalue of mf . This finishes the proof of 1) =⇒ 2) , Let us prove (2) =⇒ (1) . If λ is not an eigenvalue of mf , then mf − λ id is an invertible map, in particular it is surjective. So, there exists g ∈ P such that g(f − λ) = 1 mod I . Clearly this implies that there cannot exist p ∈ Z(I) such that f (p) − λ = 0


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The matrix MfEE If E = (t1 , . . . , tµ ) is a row of polynomials whose residue classes form a K -basis of A = P/I , then we use the short form f E to mean the row of vectors (f t1 , . . . , f tµ ) . A way of describing mf explicitly is to fix such a K -basis E of A and represent mf via the matrix MfEE where the j -column of MfEE is given by the coefficients of f tj when written in terms of E . In other words, if f tj =

µ P

akj tk

mod I

k =1

then the j th column of MfEE is (a1j , a2j , . . . , aµj )tr . A more compact way of expressing this fact is the following formula f E = E MfEE

mod I

Usually a K -basis E is an order ideal of monomials and hence we may assume that t1 = 1 . We may as well assume that the residue classes of some of the indeterminates (most of the times all) are among the entries of E . Lorenzo Robbiano (Università di Genova)

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Eigenvalues and coordinates of zeros Corollary Let xi be such that its class is among the entries of E . Then the xi coordinates of the points in ZK (I) are the eigenvalues of mxi . Proof. It follows immediately from the theorem, since xi (p) is exactly the xi coordinate of p. We observe that, as in the case of the Lex-method, we need the zeros of a polynomial (in this case the characteristic polynomial), and hence we hit again an intrinsic obstacle: we need approximation. The above corollary gives a first method for computing ZK (I) , however what we get is only a grid of points which is usually a much larger set than ZK (I) . Sometimes we can do better, but we need a description of eigenvectors. In the sequel we use the following terminology. Lorenzo Robbiano (Università di Genova)

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Eigenvectors and coordinates of zeros Definition Let ϕ : V −→ V be a linear map of K -vector spaces, let λ ∈ K be an eigenvalue of ϕ , and let Wλ = Ker(ϕ − λ id) be the corresponding eigenspace. Then the non-zero vectors in Wλ are called λ -eigenvectors, or simply eigenvectors, if the context is clear. Corollary (Eigenvectors) Let p ∈ ZK (I) , and let E = (t1 , . . . , tµ ) be a row of power products whose classes form a K -basis of A . Then the vector E(p) = (t1 (p), . . . , tµ (p)) is an f (p) -eigenvector of mf∗ . Proof. We use the formula f E = E M E fE

mod I and evaluate both sides at p . We get E f (p)E(p) = E(p) Mf E

By transposing both sides we get

tr E tr tr f (p)(E(p)) = (Mf E ) (E(p))

This formula means that E(p) is an f (p) -eigenvector of the matrix (M E )tr , hence of m∗ . f fE


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Non-derogatory matrices I

Definition A square matrix M ∈ Matµ (C) is said to be non-derogatory if the following equivalent conditions are satisfied. 1

All eigenspaces of M are 1 -dimensional.

2

The Jordan canonical form of M has one Jordan block per eigenvalue.

3

MinPolyM = CharPolyM .

Next corollary is one of the many variations which can be played around the above result. It highlights the importance of non-derogatory matrices.


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Non-derogatory matrices II Corollary Let 1, x1 , . . . , xn be the first n + 1 -entries of E , assume that the matrix (MfEE )tr is non-derogatory, let ZK (I) = Z(I) = {p1 , . . . , pr } , and let W1 , . . . , Wr be the 1 -dimensional eigenspaces corresponding to f (p1 ), . . . f (pr ) respectively. If we choose vj = (a1j , a2j , . . . , ar +1,j , . . . ) ∈ Wj \ 0 , for j = 1, . . . , r , then we have the equalities pj = (a2j /a1j , . . . , ar +1,j /a1j ) for j = 1, . . . , r. Proof. We use the above corollary about eigenvectors to deduce that the vector E(pj ) = (t1 (pj ), . . . , tµ (pj )) is an f (pj ) -eigenvector of f , hence for j = 1, . . . , r .

it is a non-zero vector in Wj

The assumption about (M E )tr means that all the Wj ’s are 1 -dimensional vector spaces, hence the vectors E(pj ) and vj fE proportional. On the other hand, we know that t1 = 1 , hence t1 (pj ) = 1 , hence a1j 6= 0 . We divide the coordinates of vj

by a1j

are

and conclude.

We observe that the computation of eigenspaces creates a new difficulty. Since the eigenvalues are in general given up to a certain degree of accuracy, the eigenspaces are not computable as kernels of linear maps. Lorenzo Robbiano (Università di Genova)

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Example I Example Let I be the ideal in P = R[x, y ] generated by the set of polynomials {x 2 + 4/3xy + 1/3y 2 − 7/3x − 5/3y + 4/3, y 3 + 10/3xy + 7/3y 2 − 4/3x − 20/3y + 4/3, xy 2 − 7/3xy − 7/3y 2 − 2/3x + 11/3y + 2/3} . We can check that this set is a DegRevLex-Gröbner basis, hence E = [1, x, y , xy , y 2 ] is a basis modulo I . By computing NF(x 2 , I) , NF(x 2 y , I) , NF(xy 2 , I) , we get 0 0 B 1 B E MxE = B B 0 @ 0 0

0

−4/3 7/3 5/3 −4/3 −1/3

0 B −4/3 B E tr 0 (MxE ) = B B @ 4/3 −2/3

1 7/3 0 −4/3 2/3

0 0 0 1 0

4/3 −4/3 4/3 1/3 −2/3

0 5/3 0 4/3 −11/3

−2/3 2/3 −11/3 7/3 7/3

0 −4/3 1 1/3 7/3

1 C C C C A

0 −1/3 0 −2/3 7/3

1 C C C C A

Its characteristic polynomial is x 5 − 5x 4 + 7x 3 + x 2 − 8x + 4 which factorizes as follows (x + 1)(x − 1)2 (x − 2)2 . Since we can check (with CoCoA) that E tr dim(W1 ) = 2 , we deduce that (MxE ) is derogatory. Lorenzo Robbiano (Università di Genova)

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Example II Example (continued) By computing NF(xy 2 , I) , NF(y 3 , I) , we get the multiplication matrix 0 0 B 0 B E MyE = B B 1 @ 0 0

0 0 0 1 0

0 0 0 0 1

−2/3 2/3 −11/3 7/3 7/3

−4/3 4/3 20/3 −10/3 −7/3

1 C C C C A

whose transposed is 0

0 B 0 B E tr 0 (MyE ) = B B @ −2/3 −4/3

0 0 0 2/3 4/3

1 0 0 −11/3 20/3

0 1 0 7/3 −10/3

0 0 1 7/3 −7/3

1 C C C C A

Its characteristic polynomial is y 5 − 5y 3 + 4y which factorizes in the following way y (y − 1)(y + 1)(y − 2)(y + 2) . We check that all the eigenspaces are 1 -dimensional, hence the matrix (M E )tr is non-derogatory. yE We may choose one non-zero vector inside each of them, for instance the vectors v1 = (1, 1, 0, 0, 0) , v2 = (1, 1, 1, 1, 1) , v3 = (1, 2, −1, −2, 1) , v4 = (1, −1, 2, −2, 4) , v5 = (1, 2, −2, −4, 4) . Therefore, five points in Z(I) are (1, 0) , (1, 1) , (2, −1) , (−1, 2) , (2, −2) . But dimR (P/I) = 5 , hence ZR (I) = Z(I) = {(1, 0), (1, 1), (2, −1), (−1, 2), (2, −2)} .


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From points to equations: the Buchberger-Möller Algorithm

History I

In the 1970s and early 1980s, algebraic geometry was essentially synonymous with Grothendieck’s scheme theory. Most geometers were busily developing new cohomology theories and proving vanishing theorems in them; some had never seen the equations of a non-trivial example for their theories. In this atmosphere the few hardy folks who dared to give talks about such down-to-earth topics as finite sets of points in affine or projective spaces were confronted with disinterest or even ridicule. Now, about a quarter of a century later, the tide has turned completely: finite sets of points are an active and well-respected branch of algebraic geometry. How did this change come about?


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History II When one starts to reduce deep problems in algebraic geometry to their essential parts, it frequently turns out that at their core lies a question which has been studied for a long time, and sometimes this question is related to finite sets of points. For instance, already in the eighteenth century G. Cramer and L. Euler discussed (in their correspondence) a phenomenon which is nowadays called the Cayley-Bacharach property. Later, in 1843, A. Cayley formulated a theorem which was a vast generalization of what Cramer and Euler had stumbled upon. But unfortunately the claim was false and the proof invalid. This was not corrected until 1886, when I. Bacharach used M. Noether’s “ AΦ + BΨ ”-theorem to give a correct statement and a true proof. Many decades later it turned out that the Cayley-Bacharach property is connected to the “Gorenstein property” of the homogeneous coordinate ring of a projective point set, and today a generalization of this property is central to an important conjecture by D. Eisenbud, M. Green and J. Harris. Lorenzo Robbiano (Università di Genova)

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History III

What does all of this have to do with Computational Commutative Algebra? We think that finite sets of points provide excellent examples for the ways in which computer algebra methods can be applied. They show up in many branches of mathematics besides algebraic geometry, for instance in interpolation, coding theory, and statistics. Efficient algorithms help us to compute with larger and larger point sets, to check results and conjectures, and to discover new ones. And today we have a new entry in the game: ideals of finite sets of points with approximate coordinates.


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Separators and Interpolators

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 ) . 1

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 6= i .

2

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 .


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Separators exist

It is clear that 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) . Proposition Let X = {p1 , . . . , ps } ⊆ K n be an affine point set, and let X be the tuple (p1 , . . . , ps ) . 1

For every i ∈ {1, . . . , s} , there exists a separator of pi from X \ pi .

2

For every (a1 , . . . , as ) ∈ K s , there exists an interpolator for (a1 , . . . , as ) at X .

3

Let Y be an affine point set contained in X .P For every pi ∈ X , let fi ∈ P be a separator of pi from X \ pi , and let fX\Y = pi ∈X\Y fi . Then we have I(Y) = I(X) + (fX\Y ) .


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The BM Algorithm I 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 )) ∈ K s 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 − from L all multiples of t . Then continue with step 2).

P i ai si

to G where si

is the i th element in S . Remove

P (5) Otherwise (v1 , . . . , vs ) 6= (0, . . . , 0) , so append (v1 , . . . , vs ) as a new row to M and t − i ai si as a new element to S . Add t to O , and add to L those elements of {x1 t, . . . , xn t} 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)} . Lorenzo Robbiano (Università di Genova)

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The BM Algorithm II A small alteration of this algorithm allows us to compute the separators of X as well. Corollary In the setting of the theorem, replace step 2) by the following instruction. (2’) If L = ∅ then row reduce M to a diagonal matrix and mimic these row operations on the elements of S (considered as a column vector). Next replace S by M−1 S , return the triple (G, O, S) , and stop. If L 6= ∅ , choose the term t = minσ (L) and delete it from L . The resulting sequence of instructions defines again an algorithm. It returns a triple (G, O, S) such that G is the reduced σ -Gröbner basis of I(X) , such that O = Tn \ LTσ {I(X)} , and the tuple S contains the separators of pi from X \ pi for i = 1, . . . , s .


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Remarks

More details on BM can be found in the two papers (see [ABKR] and [AKR]). Using Buchberger-Möller algorithm and its corollary it is possible to compute ideals of points and to solve the problem of interpolation. What happens if the data are not exact? Approximate versions of the above results should be able to construct sets of polynomials which almost vanish at X , almost separators and almost interpolators. Of course new problems of stability arise in this context.


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Border Bases

Bases of P/I In the last section we pointed to questions of stability. We have seen that methods for solving polynomial systems inevitably feature problems of approximation. On the other hand, problems of the same type arise if the data are not exact. If we go back for a moment to the eigenvalue method, we remember that it requires multiplication matrices which, in turn, require the choice of a basis of A = P/I as a K -vector space. One way to get a basis of A is via Gröbner bases. If σ is a term ordering on Tn and G = {f1 , . . . , fs } is a σ -Gröbner basis of 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 . Therefore, once a σ -Gröbner basis of I is computed, a basis is available. If we change σ we may get different bases of A , and a first question arises here. Question 1 Are these the only bases? And in connection with the stability issue, the next question is. Question 2 What are the most stable bases? Lorenzo Robbiano (Università di Genova)

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Border Bases

Filtration cycle I Before answering these question, we make a digression into the realm of statistics (see Tutorial 92 of [KR2]). A classical real world problem reads as follows. A number of similar chemical plants had been successfully operating for several years in different locations. In a newly constructed plant the filtration cycle took almost twice as long as in the older plants. Seven possible causes of the difficulty were considered by the experts. 1 2

3

4 5 6 7

The water for the new plant was different in mineral content. The raw material was not identical in all respects to that used in the older plants. The temperature of filtration in the new plant was slightly lower than in the older plants. A new recycle device was absent in the older plants. The rate of addition of caustic soda was higher in the new plant. A new type of filter cloth was being used in the new plant. The holdup time was lower than in the older plants.


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Border Bases

Filtration cycle II

These causes lead to seven variables x1 , . . . , x7 . Each of them can assume only two values, namely old and new which we denote by 0 and 1, respectively. All possible combinations of these values form the full design D = {0, 1}7 ⊆ A7 (Q) . Its vanishing ideal is I(D) = (x12 − x1 , x22 − x2 , . . . , x72 − x7 ) in the polynomial ring Q[x1 , . . . , x7 ] . Our task is to identify an unknown function ¯f : D −→ K , namely the length of a filtration cycle. This function is called the model, since it is a mathematical model of the quantity which has to be computed or optimized.


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Border Bases

Filtration cycle III In order to fully identify it, we would have to perform 128 = 27 cycles. This is impracticable since it would require too much time and money. On the other hand, suppose for a moment that we had conducted all experiments and the result was ¯f = a + b x1 + c x2 for some a, b, c ∈ Q . At this point it becomes clear that we have wasted many resources. Had we known in advance that ¯f is given by a polynomial having only three unknown coefficients, we could have identified them by performing only three suitable experiments! Namely, if we determine three values of a + b x1 + c x2 and the associated matrix of coefficients is invertible, we can easily find a, b, c by solving a system of three linear equations in these three unknowns. However, a priori one does not know that the answer has the shape indicated above. In practice, one has to make some guesses, perform well-chosen experiments, and possibly modify the guesses until the process yields the desired answer. In the case of the chemical plant, it turned out that only x1 and x5 were relevant for identifying the model. Lorenzo Robbiano (Università di Genova)

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Border Bases

Fractions Motivated by this example, we introduce a piece of notation. Let K be a field. For i = 1, . . . , n , let ì ≥ 1 and Di = {ai1 , ai2 , . . . , aiì } ⊆ K . Then we say that the full design D = D1 × · · · × Dn ⊆ An (K ) has levels (`1 , . . . `n ) . The polynomials fi = (xi − ai1 ) · · · (xi − aiì ) with i = 1, . . . , n generate the vanishing ideal I(D) ⊆ P of D . They are called the canonical polynomials of D . For any term ordering σ on Tn , the canonical polynomials are the reduced σ -Gröbner basis of I(D) . Thus the order ideal OD = {x1α1 · · · xnαn | 0 ≤ αi < ì for i = 1, . . . , n} is canonically associated to D and represents a K -basis of P/I(D) . Our main task is to identify an unknown function ¯f : D −→ K called the model. We want to choose a fraction F ⊆ D that allows us to identify the model if we have some extra knowledge about the form of ¯f . Lorenzo Robbiano (Università di Genova)

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Border Bases

Inverse problem

Given a full design D and an order ideal O ⊆ OD , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P/I(F ) ? It is called the inverse problem in DoE (Design of Experiments). The solution of this problem already in year 2001 (see [CR]) showed the importance of a new notion in commutative algebra, that of border basis. It was a first answer to Question 1. Let us move to Question 2 by considering the following example.


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Border Bases

Two conics I

Example Consider the polynomial system f1 f2

= =

1 4

x2 + y2 − 1 x + 14 y 2 − 1 2

= 0 = 0

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

p

p 4/5, ± 4/5)} .

The set {x 2 − 45 , y 2 − 45 } is the reduced Gröbner basis of the ideal I = (f1 , f2 ) ⊆ C[x, y ] with respect to σ = DegRevLex . Therefore we have LTσ (I) = (x 2 , y 2 ) , 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)

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Border Bases

Two conics II

Now consider the slightly perturbed polynomial system ˜f1 ˜f2

= 14 x 2 + y 2 + ε xy − 1 = = x 2 + 14 y 2 + ε xy − 1 =

0 0

where ε is a small number. e The intersection of Z(˜f1 ) and Z(˜f2 ) consists of four perturbed points X close to those in X . Lorenzo Robbiano (Università di Genova)

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Border Bases

Two conics III

This time the ideal ˜I = (˜f1 , ˜f2 ) has the reduced σ -Gröbner basis {x 2 − y 2 , xy +

5 4ε

y 2 − 1ε , y 3 −

16ε 16ε2 −25

x+

20 16ε2 −25

y}

Moreover, we have LTσ (˜I) = (x 2 , xy , y 3 ) and T2 \ LTσ {˜I} = {1, x, y , y 2 } . 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 of 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.


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Border Bases

BB: Introduction I 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 of P/I and by the multiplication table of this basis. We can build border basis theory in the following way. Let O = {t1 , . . . , tµ } be an order ideal and ∂O = {b1 , . . . , bν } its P border. A µ border prebasis consists of polynomials of the form gj = bj − i=1 αij ti with αij ∈ K and can be used for border division. It is a border basis if and only if the residue classes of {t1 , . . . , tµ } form a K -basis of P/I . If O represents a vector space basis of P/I , then an O -border basis of I exists, is uniquely determined, and generates the ideal. Furthermore, it is then possible to define and compute normal forms with respect to O .


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Border Bases

BB: Introduction II Let K be a field, let P = K [x1 , . . . , xn ] be standard graded, and let Tn be the monoid of terms in P . Definition Let O be a non-empty subset of Tn . 1

2

The closure of O is the set O of all terms in Tn which divide one of the terms of O . The set O is called an order ideal if O = O , i.e. O is closed under forming divisors.

Definition A set of polynomials G = {g1 , . . . , gν } in P P is called an O -border prebasis µ if the polynomials have the form gj = bj − i=1 αij ti with αij ∈ K for 1 ≤ i ≤ µ and 1 ≤ j ≤ ν .


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Border Bases

BB: Introduction III

Definition Let G = {g1 , . . . , gν } be an O -border prebasis, let G be the tuple (g1 , . . . , gν ) , and let I ⊆ P be an ideal containing G . The set G or the tuple G is called an O -border basis of I if one of the following equivalent conditions is satisfied. 1 The residue classes O = {¯t1 , . . . , ¯tµ } form a K -vector space basis of P/I . 2

We have I ∩ hOiK = {0} .

3

We have P = I ⊕ hOiK .


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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 . 1

There exists a unique O -border basis of I .

2

Let G be an O -border prebasis whose elements are in I . Then G is the O -border basis of I .

3

Let k be the field of definition of I . Then the O -border basis of I is contained in k [x1 , . . . , xn ] .

Proposition 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 of Oσ (I) . Lorenzo Robbiano (Università di Genova)

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Border Bases

Existence and uniqueness Border bases can be characterized imitating Gröbner bases theory. So we can use special generation, rewrite relations, syzygies and an important new feature.

Definition Pµ Let G = {g1 , . . . , gν } be an O -border prebasis, i.e. let gj = bj − i=1 αij ti with αij ∈ K for i = 1, . . . , µ and j = 1, . . . , ν . Given r ∈ {1, . . . , n} , we (r ) define the r th formal multiplication matrix Xr = (ξk ` ) of G by ( δki , if xr t` = ti (r ) ξk ` = αkj , if xr t` = bj Here we let δki = 1 if k = i and δki = 0 otherwise.


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Border Bases

Commuting matrices

The following is a fundamental fact. 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. In that case the formal multiplication matrices represent the multiplication endomorphisms of P/I with respect to the basis {¯t1 , . . . , ¯tµ } .


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Border Bases

Two conics IV What are the border bases in the two cases of the conics and the perturbed conics?

Two conics {x 2 − 45 ,

x 2 y − 45 y ,

xy 2 − 45 x,

y 2 − 45 }

Two perturbed conics {x 2 +

4 5

xy 2 +

20 16ε2 −25


x 2y −

εxy − 45 , x−

16ε 16ε2 −25

y,

Polynomial Models

16ε x 16ε2 −25 2

20 y, 16ε2 −25 4 4 5 εxy − 5 }

+

y +


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Border Bases

BB Algorithm The most important feature of border bases is, of course, that they can also be computed. Theorem (The Border Basis Algorithm) Let I ⊆ P be a zero-dimensional ideal generated by a set of non-zero polynomials {f1 , . . . , fs } , and let σ be a degree compatible term ordering. Consider the following sequence of instructions. 1 2

Let V0 ⊆ P be the K -vector subspace generated by {f1 , . . . , fs } . Let d = max{deg(t) | t ∈ Supp(f1 ) ∪ · · · ∪ Supp(fs )} and L = Tn . ≤d

3

For i = 0, 1, 2, . . .

4

Compute an order ideal O ⊆ L such that the residue classes of the terms in O form a K -vector space basis of hLiK /Vi .

5

Check whether ∂O ⊆ L . If this is not the case, increase d by one, replace L by Tn , replace V0 by Vi , and continue with ≤d step 3).

6

Let O = {t1 , . . . , tµ } and ∂O = {b1 , . . . , bν } . For j = 1, . . . , ν , use linear algebra to compute the representation ¯ = Pµ α ¯t of b ¯ ∈ hLi /V in terms of the basis {¯t , . . . , ¯tµ } and let g = b − Pµ α t . Then let b j 1 j j j K i i=1 ij i i=1 ij i G = {g1 , . . . , gν } , return the pair (O, G) , and stop.

compute Vi+1 = (Vi + x1 Vi + · · · + xn Vi ) ∩ hLi K

until Vi+1 = Vi .

This is an algorithm which returns a pair (O, G) where O is an order ideal and G is the O -border basis of I . Lorenzo Robbiano (Università di Genova)

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Giacimenti petroliferi I

Un giacimento petrolifero non è paragonabile ad un lago sotterraneo di petrolio. Se così fosse, non sarebbe difficile estrarre tutto il petrolio disponibile, anche se si trovasse a enormi profondità. Nelle aree che producono petrolio e gas la parte più superficiale della crosta terrestre è fatta di rocce sedimentarie. Il gas è più leggero del petrolio, il petrolio è più leggero dell’acqua e la porosità della roccia permette alle suddette sostanze di salire verso la superficie terrestre. Se in questo processo esse incontrano una trappola, si distribuiscono secondo la loro densità, quindi il gas si troverà in alto, a metà troveremo il petrolio e più in basso l’acqua, sempre distribuiti in minuscole gocciole che permeano le porosità della roccia.


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Giacimenti petroliferi II I popoli dell’antichità conoscevano alcuni giacimenti di petrolio superficiali, che utilizzavano per produrre medicinali e bitume e successivamente per alimentare le lampade. Anche i primi minatori all’inizio dell’era industriale ebbero qualche successo, dato che alcune trappole non erano del tutto sigillate. Nel ventesimo secolo, ci si rese conto che era necessario studiare la struttura geologica delle rocce e fare dei fori di prova. Se si ha fortuna, un lungo tubo metallico si immette nel foro; nasce così un pozzo petrolifero. Mentre il gas fluisce spontaneamente verso la superficie, solo in qualche caso c’è nel sottosuolo pressione sufficiente a far fluire attraverso i tubi anche il petrolio. La maggior parte dei pozzi invece non ha abbastanza pressione e quindi nei tubi di produzione si deve immettere gas, il quale si mescola al petrolio, lo rende più leggero e gli permette la fuoriuscita. Lorenzo Robbiano (Università di Genova)

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Giacimenti petroliferi III Una volta giunti in superficie, i fluidi sono trasportati tramite tubi ad un recipiente, detto separatore, dove i tre costituenti, petrolio, acqua e gas, vengono separati. Durante lo sfruttamento di un giacimento, la pressione dei fluidi varia, in generale diminuisce. Ad un certo punto non si riesce più ad estrarre petrolio e il giacimento viene abbandonato. Perché è difficile studiare e ottimizzare la produzione? Innanzitutto va detto che l’osservazione diretta dello stato fisico-chimico di un giacimento non è possibile. Simulazioni in laboratorio sono, appunto, simulazioni; chi può dire quello che realmente avviene nelle profondità della crosta terrestre? Non potrebbero intervenire fenomeni governati da leggi fisiche sconosciute?


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Giacimenti petroliferi IV Gli studi di laboratorio hanno portato alla seguente situazione: quando un giacimento viene abbandonato si stima che circa il 20% del gas sia andato perso, ma soprattutto... che circa il 70% di petrolio sia andato perso!!!. Perché succede ciò? E quale è la soluzione del problema? Alla seconda domanda alcuni governi rispondono con una dichiarazione di guerra a paesi ricchi di petrolio. Si può anche rispondere dicendo che sarebbe opportuno diminuire l’uso del petrolio, soluzione decisamente migliore di quella precedente. In ogni caso sarebbe utile dare una risposta alla prima domanda. Perché dunque si perde circa il 70% del petrolio di un giacimento? A detta degli esperti, i pozzi petroliferi sono dei gioielli di ingegneria e probabilmente non sarebbe facile migliorarne l’efficienza in modo sostanziale. Lorenzo Robbiano (Università di Genova)

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Giacimenti petroliferi V

Ma in un giacimento sono presenti molti pozzi e ogni singolo pozzo può trarre fluidi da zone differenti. In questi casi il vero punto dolente è il fatto che non è assolutamente chiaro come l’estrazione dei fluidi dai diversi pozzi influenzi l’estrazione di ogni singolo pozzo. E anche nel caso più semplice di un pozzo che riceva fluidi da zone diverse, l’interazione non è affatto chiara.

Che fare?


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A two-zone well


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The indeterminates The first step is to associate the indeterminates xi with physical quantities in the production problem. In the sequel we use n = 5 and the following associations, where the physical quantities are the ones referenced in the previous figure.

x1

:

∆Pinflow 1

x2

:

∆Pinflow 2

x3

: Gas production

x4

:

∆Ptub

x5

:

∆Ptransport

Table: Physical interpretation of the indeterminates.


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Oil production The physical quantities listed in the above table may all be interpreted as driving forces for the oil production. For the pressure differences ∆P this is clear. But it holds also for the gas production. When a large amount of gas is produced in the deeper parts of the reservoir, it disperses in the fluid mixture, makes it lighter, and in this way stimulates oil production through this lifting process. Thus the physical quantities listed in the above table may all be viewed as the causing quantities, or inputs, and the oil production is their effect, or output. Find a polynomial f ∈ R[x1 , . . . , x5 ] , using only the evaluations X of the quantities xi and the evaluations of f !


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Errors Notice also that all data are based on measurements, i.e. they may contain measurement errors. Consequently, we can only expect that the desired polynomial f vanishes approximately at the points of X . We say that a polynomial f ∈ R[x1 , . . . , xn ] vanishes ε -approximately at X if |f (pi )| < ε for i = 1, . . . , s . This definition entails several problems. 1 2

The polynomials which vanish ε -approximately at X do not form an ideal! All polynomials with very small coefficients vanish ε -approximately at X !

Definition An ideal I ⊆ P is called an ε -approximate vanishing ideal of X if there exists a system of generators {f1 , . . . , fr } of I such that kfi k = 1 and fi vanishes ε -approximately at X for i = 1, . . . , r .


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A Polynomial Model

The calculation is done using the AVI algorithm with ε = 0.1 and term ordering DegRevLex .

The result is a polynomial g of the form g

=

−5.35y3 y52 − 0.73y4 y52 − 0.21y53 + 2.37y2 y3 −7.32y32 − 0.88y1 y4 − 0.15y2 y4 + 0.34y3 y4 −0.55y42 − 2.20y1 y5 − 0.35y2 y5 + 3.85y3 y5 +0.67y4 y5 + 0.61y52 + 0.62y1 − 0.26y2 +2.69y3 + 0.98y4 + 1.63y5 − 0.12


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A Picture of the Model

Figure: Model construction using the AVI algorithm.


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Thinning Out and The SOI Algorithm

Definition Let O be an order ideal of Tn . The set O is called stable w.r.t. Xε if the e evaluation matrix evalXe (O) has full rank for each admissible perturbation X of X . Furthermore, if O is also a basis of P/I(X) , it is called a stable quotient basis of I(X) .

Theorem Let X be a finite set of points in Rn and O a quotient basis for I(X) . Then there exists a tolerance δ = (δ1 , . . . , δn ) , with δi > 0 , such that O is stable w.r.t. Xδ .


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Thinning Out and The SOI Algorithm Theorem Let Xε be a set of s distinct empirical points, and let O = {t1 , . . . , ts } be a quotient basis for I(X) which is stable w.r.t. Xε . Then, for each admissible e has an O -border e of Xε , the vanishing ideal I(X) perturbation X e basis G .

The theorem leads to the SOI Algorithm, which is very useful for theoretical reasons. A more efficient variant is the NBM Algorithm. Both take advantage of a special technique used to thin out the data.


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The three points 1

We want to represent all zero-dimensional subschemes of A2K such that the residue classes of the elements in {1, x, y } form a basis of their coordinate ring, when viewed as a K -vector space. In particular, the elements x 2 , xy , y 2 should be expressed modulo I as linear combination of the elements in {1, x, y } . In other words, the ideal I must contain three polynomials g1 = x 2 − c11 − c21 x − c31 y , g2 = xy − c12 − c22 x − c32 y , g3 = y 2 − c13 − c23 x − c33 y for suitable values of the coefficients cij . Lorenzo Robbiano (Università di Genova)

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The three points 2 {1, x, y } is an order ideal of monomials and that the complementary monomial ideal is generated by {x 2 , xy , y 2 } . If σ is a degree-compatible term ordering, for instance σ = DegRevLex then LTσ (g1 ) = x 2 , LTσ (g2 ) = xy , LTσ (g3 ) = y 2 , no matter which values are taken by the cij ’s. We know that dimK (P/I) = dimK (P/ LTσ (I)) and we want that this number is 3 . On the other hand, dimK (P/(x 2 , xy , y 2 )) = 3 , so we want that LTσ (I) = (x 2 , xy , y 2 ) . In other words we want to impose that {g1 , g2 , g3 } is a σ -Gröbner basis of I . How do we do that? There are two fundamental syzygies of the power products x 2 , xy , y 2 namely (−y , x, 0) and (0, −y , x) .


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The three points 3 Imposing that {g1 , g2 , g3 } is the reduced σ -Gröbner basis of I is equivalent to imposing that some polynomial expressions in the cij are zero. Let J be the ideal of K [c11 , . . . , c33 ] generated by these polynomials. We check with CoCoA that there is an isomorphism between K [c11 , . . . , c33 ]/J and K [c21 , c31 , c22 , c32 , c23 , c33 ] . The conclusion is that our family is parametrized by an affine space A6K . All such ideals are generated by {g1 , g2 , g3 } where 2 g1 = x 2 − (−c21 c32 − c31 c33 + c22 c31 + c32 ) − c21 x − c31 y

g2 = xy − (c21 c22 + c31 c23 − c22 c21 − c32 c22 ) − c22 x − c32 y 2 g3 = y 2 − (c22 + c32 c23 − c23 c21 − c33 c22 ) − c23 x − c33 y

and the parameters can vary freely. Q1: Consider the process of imposing that {g1 , g2 , g3 } is the reduced σ -Gröbner basis of I . Is it canonical? Q2: How do we explain that the dimension of the parameter space is 6? Lorenzo Robbiano (Università di Genova)

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Border Basis Scheme The following material is taken from [KR3]. Let O = {t1 , . . . , tµ } be an order ideal in Tn , and let ∂O = {b1 , . . . , bν } be its border. We define a moduli space 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 X

cij ti

i=1 2

3

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


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Universal family Definition Let g1 , . . . , gν ∈ K [x1 , . . . , xn , c11 , . . . , cµν ] , with gj = bj − for j = 1, . . . , ν , be the generic O -border prebasis.

Pµ

i=1

cij ti

Let BO = K [c11 , . . . , cµν ]/I(BO ) , and let UO = K [x1 , . . . , xn , c11 , . . . , cµν ]/ I(BO ) + (g1 , . . . , gν ) . Then the natural homomorphism of K -algebras Φ : BO −→ UO is called the universal O -border basis family. The fibers of the basis family are the quotient rings P/I where I is a zero-dimensional ideal which has an O -border basis. Theorem (The Free Family) Let BO , UO , and Φ : BO −→ UO be as above. The residue classes of the elements in O are a BO -module basis of UO . The map Φ is a flat homomorphism. Lorenzo Robbiano (Università di Genova)

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Philosophy In the paper [KR3] we study properties of the border basis scheme. For instance we show that in some cases it is an affine space. Suppose we are computing the border basis of an ideal of points I in P . We have 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. Consequently, the border basis is modified slightly into a new border basis of a zero-dimensional ideal. This sentence can be rephrased by saying that moving the points moves the border basis, and the movement traces a path inside the border scheme. On the other hand, if we perturb the equations of the border basis, we are not at all guaranteed that we keep inside the border scheme. What we can say is that the multiplication matrices almost commute, but most likely the new ideal is the unit ideal. This is an important remark since rarely the border basis scheme is an affine space. One philosophical question here is the following. When we compute with approximate data, do we want to move along the border basis scheme, or are we content to simply stay close to the border basis scheme? Lorenzo Robbiano (Università di Genova)

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Four Points: the Gröbner Viewpoint I Example Let O = {1, x, y , xy } . We observe that t1 = 1 , t2 = x , t3 = y , t4 = xy , b1 = x 2 , b2 = y 2 , b3 = x 2 y , b4 = xy 2 . Let σ = DegRevLex , so that x >σ y . g1 g2 g3 g4

= x 2 − c11 1 − c21 x − c31 y − c41 xy = y 2 − c12 1 − c22 x − c32 y − c42 xy = x 2 y − c13 1 − c23 x − c33 y − c43 xy = xy 2 − c14 1 − c24 x − c34 y − c44 xy

Then necessarily c42 = 0 (the linear space), so that g2 is replaced by g2∗ = y 2 − c12 1 − c22 x − c32 y Then {g1 , g2 } is the reduced Gröbner basis. There are 7 free parameters.


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Four Points: the Gröbner Viewpoint II

Example To make everything compatible with σ we need x > y , y 2 > 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, . . . This method works in general (see [R]).


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Four Points: the Border Viewpoint I

The generic multiplication matrices are 0 c 0 c14 12 1 c22 0 c24 Ax = and Ay = 0 c32 0 c34 0

c42

1

c44

0 0 1 0

0 0 0 1

c11 c21 c31 c41

c13 c23 c33 c43

The ideal I(BO ) is generated by {c11 c32 + c13 c42 − c14 , c12 c21 + c14 c41 − c13 , c12 c23 − c11 c34 + c14 c43 − c13 c44 , c21 c32 + c23 c42 − c24 , c21 c22 + c24 c41 + c11 − c23 , c22 c23 − c21 c34 + c24 c43 − c23 c44 + c13 , c31 c32 + c33 c42 + c12 − c34 , c21 c32 + c34 c41 − c33 , c23 c32 − c31 c34 + c34 c43 − c33 c44 − c14 , c32 c41 + c42 c43 + c22 − c44 , c21 c42 + c41 c44 + c31 − c43 , c34 c41 − c23 c42 + c24 − c33 }


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Four Points: the Border Viewpoint II

The ideal I(BO ) can also be generated by {c23 c41 c42 − c21 c42 c43 + c21 c44 + c11 − c23 , c34 c41 c42 − c32 c41 c44 + c32 c43 + c12 − c34 , −c23 c32 c41 + c21 c32 c43 − c21 c34 + c13 , −c21 c34 c42 + c21 c32 c44 − c23 c32 + c14 , c32 c41 + c42 c43 + c22 − c44 , −c21 c32 − c23 c42 + c24 , c21 c42 + c41 c44 + c31 − c43 , −c21 c32 − c34 c41 + c33 } We see from the second representation that there are 8 free indeterminates, namely c21 , c23 , c32 , c34 , c41 , c42 , c43 , c44 while the remaining indeterminates depend on the free ones by the polynomial expressions above.


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Four Points: the Border Viewpoint III We can then conclude that the border scheme BO is an affine cell (of the corresponding Hilbert scheme) explicitly represented by the isomorphism BO −→ K [c21 , c23 , c32 , c34 , c41 , c42 , c43 , c44 ] given by c11 c12 c13 c14 c22 c24 c31 c33


−→ −c23 c41 c42 + c21 c42 c43 − c21 c44 + c23 −→ −c34 c41 c42 + c32 c41 c44 − c32 c43 + c34 −→ c23 c32 c41 − c21 c32 c43 + c21 c34 −→ c21 c34 c42 − c21 c32 c44 + c23 c32 −→ −c32 c41 − c42 c43 + c44 −→ c21 c32 + c23 c42 −→ −c21 c42 − c41 c44 + c43 −→ c21 c32 + c34 c41

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Four Points: the Border Viewpoint IV

Such isomorphism induces an isomorphism of UO with the ring K [x, y , c21 , c23 , c32 , c34 , c41 , c42 , c43 , c44 ]/(ge1 , ge2 , ge3 , ge4 ) where 2 f g 1 = y − (−c23 c41 c42 + c21 c42 c43 − c21 c44 + c23 ) − c21 x − (−c21 c42 − c41 c44 + c43 )y − c41 xy , 2 f g 2 = x − (−c34 c41 c42 + c32 c41 c44 − c32 c43 + c34 ) − (−c32 c41 − c42 c43 + c44 )x − c32 y − c42 xy , 2 f g 3 = xy − (c23 c32 c41 − c21 c32 c43 + c21 c34 ) − c23 x − (c21 c32 + c34 c41 )y − c43 xy , 2 f g 4 = x y − (c21 c34 c42 − c21 c32 c44 + c23 c32 ) − (c21 c32 + c23 c42 )x − c34 y − c44 xy ,

The ideal (ge1 , ge2 , ge3 , ge4 ) can be considered as the ideal of all the subschemes of the affine plane of length 4 whose coordinate ring admits O as a K -vector space basis.


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Four Points: the Border Viewpoint V The border basis scheme is an affine cell in the Hilbert Scheme, so we can construct a flat deformation (along a rational curve) which connects all the ideals in the family to the unique monomial ideal (x 2 , y 2 ) . It suffices to substitute each free indeterminate cij with tcij where t is a parameter. We get the family of ideals ΦK [t] : K [t] −→ K [x, y , t, c21 , c23 , c32 , c34 , c41 , c42 , c43 , c44 ]/(g1 , g2 , g3 , g4 ) where g1 = y 2 − (−t 3 c23 c41 c42 + t 3 c21 c42 c43 − t 2 c21 c44 + tc23 ) − tc21 x − (−t 2 c21 c42 − t 2 c41 c44 + tc43 )y − tc41 xy , g2 = x 2 − (−t 3 c34 c41 c42 + t 3 c32 c41 c44 − t 2 c32 c43 + tc34 ) − (−t 2 c32 c41 − t 2 c42 c43 + tc44 )x − tc32 y − tc42 xy , g3 = xy 2 − (t 3 c23 c32 c41 − t 3 c21 c32 c43 + t 2 c21 c34 ) − tc23 x − (t 2 c21 c32 + t 2 c34 c41 )y − tc43 xy , g4 = x 2 y − (t 3 c21 c34 c42 − t 3 c21 c32 c44 + t 2 c23 c32 ) − (t 2 c21 c32 + t 2 c23 c42 )x − tc34 y − tc44 xy ,

It is flat and connects the generic ideal to the border ideal (y 2 , x 2 , xy 2 , x 2 y ) = (x 2 , y 2 ) . deformations simply do not exist (Pablo Picasso) Lorenzo Robbiano (Università di Genova)

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References

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, M. Kreuzer, L. Robbiano Computing zero-dimensional Schemes J. Symb. Comput. Vol 39, pp 31–49 (2005). J. Abbott, C. Fassino, M. Torrente, Thinning out redundant empirical data Preprint, arXiv.org:math/0702.327 (2007) J. Abbott, C. Fassino, M. Torrente, Notes on Stable Border Bases Preprint, arXiv.org:math/0706.231 (2007) M. Caboara and L. Robbiano Families of Estimable Terms Proc. ISSAC ’01, 56–63 (2001) The CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.


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References

References II D. Cox, Solving equations via algebras, in: A. Dickenstein and I. Emiris (eds.), Solving Polynomial Equations, Springer, Berlin (2005). C. Fassino, An approximation of the Gröbner basis of ideals of perturbed points, part I Preprint, arXiv.org:math/0703154 (2007). D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse, Approximate Computation of Zero-Dimensional Polynomial Ideals, Preprint (2006). M. Kreuzer, H. Poulisse and L. Robbiano From Oil Fields to Hilbert Schemes, Preprint (2009). M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer, Heidelberg (2000). M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer, Heidelberg (2005).


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References

References III

M. Kreuzer and L. Robbiano, Deformations of Border Bases, Collectanea Math. Vol 59, pp 275–297 (2008). L. Robbiano, On border basis and Gröbner basis schemes, Collectanea Math. Vol 60 pp 11–25 (2009). 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).


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Polynomial Models: from data to equations, from equations to solutions

Polynomial Models: from data to equations, from equations to solutions

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