From Oil Fields to Hilbert Schemes Lorenzo Robbiano Universit di - PowerPoint PPT Presentation

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 Interpolation on Finite Sets of Points 1 Data Affected by Errors 2 Border Bases 3 Families of Border Bases 4 Gröbner and Border Basis Schemes 5 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 = { p 1 , . . . , p s } ⊆ K n be an affine point set, and let X be the tuple ( p 1 , . . . , p s ) . Let i ∈ { 1 , . . . , s } . A polynomial f ∈ P is called a separator of p i from X \ p i if f ( p i ) = 1 and f ( p j ) = 0 for j � = i . Let a 1 , . . . , a s ∈ K . A polynomial f ∈ P is called an interpolator for the tuple ( a 1 , . . . , a s ) at X if f ( p i ) = a i 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 = { p 1 , . . . , p s } ⊆ K n be an affine point set, and let X be the tuple ( p 1 , . . . , p s ) . For every i ∈ { 1 , . . . , s } , there exists a separator of p i from X \ p i . For every ( a 1 , . . . , a s ) ∈ K s , there exists an interpolator for ( a 1 , . . . , a s ) 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 T n , and let X = { p 1 , . . . , p s } be an affine point set in K n whose points p i = ( c i 1 , . . . , c in ) are given via their coordinates c ij ∈ K . Consider the following sequence of instructions. 1) Let G = ∅ , O = ∅ , S = ∅ , L = { 1 } , and let M = ( mij ) ∈ Mat 0 , s ( K ) be a matrix having s columns and initially zero rows. If L = ∅ , return the pair ( G , O ) and stop. Otherwise, choose the term t = min σ ( L ) and delete it from L . 2) Compute the evaluation vector ( t ( p 1 ) , . . . , t ( ps )) ∈ Ks and reduce it against the rows of M to obtain 3) ( v 1 , . . . , vs ) = ( t ( p 1 ) , . . . , t ( ps )) − P ai ( mi 1 , . . . , mis ) with ai ∈ K i i ai si to G where si is the i th element in S . Remove from L all 4) If ( v 1 , . . . , vs ) = ( 0 , . . . , 0 ) then append the polynomial t − P multiples of t . Then continue with step 2). 5) Otherwise ( v 1 , . . . , vs ) � = ( 0 , . . . , 0 ) , so append ( v 1 , . . . , 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 { x 1 t , . . . , xnt } which are neither multiples of an element of L nor of LT σ ( 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 = T n \ LT σ {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 p i and two interpolators for a tuple ( a 1 , . . . , a s ) ∈ 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 T n and G = { f 1 , . . . , f s } is a σ -Gröbner basis of I , then LT σ { I } = { LT σ ( f 1 ) , . . . , LT σ ( f s ) } . We know that the residue classes of the elements of T n \ LT σ { 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 4 x 2 + y 2 − 1 1 f 1 = = 0 x 2 + 1 4 y 2 − 1 = = f 2 0 � � X = Z ( f 1 ) ∩ Z ( f 2 ) consists of the four points X = { ( ± 4 / 5 , ± 4 / 5 ) } . The set { x 2 − 4 5 , y 2 − 4 5 } is the reduced Gröbner basis of the ideal I = ( f 1 , f 2 ) ⊆ C [ x , y ] with respect to σ = DegRevLex . Therefore we have LT σ ( I ) = ( x 2 , y 2 ) , and the residue classes of the terms in T 2 \ LT σ { 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 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ f 2 = = 0 where ε is a small number. The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) 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 = (˜ f 1 , ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } Moreover, we have LT σ (˜ I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ I } = { 1 , x , y , y 2 } . A small change in the coefficients of f 1 and f 2 has led to a big change in the Gröbner basis of ( f 1 , f 2 ) and in the associated vector space basis of C [ x , y ] / ( f 1 , f 2 ) , 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