On Symbolic Computation of Ideal Projectors and ...

On Symbolic Computation of Ideal Projectors and Inverse Systems Boris Shekhtman

Abstract A zero-dimensional ideal J in the ring k[x] of polynomials in d variables is often given in terms of its “border basis”; that is a particular finite set of polynomials that generate the ideal. We produce a convenient formula for symbolic computation of the space of functionals on k[x] that annihilate J. The formula is particularly useful for computing an explicit form of an ideal projector from its values on a certain finite set of polynomials.

1 Introduction Throughout, k will stand for the field of complex numbers or the field of real numbers, k[x] := k [x1 , . . . , xd ] will denote the space (algebra, ring) of polynomials in d indeterminants with coefficients in the field k and (k[x])0 is the algebraic dual of k[x], i.e., the space of all linear functionals on k[x]. Definition 1 ([1]). A linear idempotent operator P : k[x] → k[x] is called an ideal projector if ker P is an ideal in k[x]. Lagrange interpolation projectors, Taylor projectors and, in one variable, Hermite interpolation projectors are all examples of ideal projectors. Thus the study of ideal projectors holds a promise of an elegant extension of operators, traditionally used in approximation theory, to multivariate setting. The theory was initiated by G. Birkhoff [1], Carl de Boor [3], C. de Boor and A. Ron [4], H. M. M˝oeller [7] and T. Sauer [10]. Any finite-dimensional projector, ideal or not, can be written as P f = ∑ λ j ( f )g j

(1)

Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, e-mail: [email protected]

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Boris Shekhtman

where (g j ) ⊂ k[x] is a (linear) basis for for the range of P and (λ j ) ⊂ (k[x])0 are dual functionals: λk (g j ) = δk, j (2) forming a basis in the space ran P∗ = (ker P)⊥ Thus the functionals (λ j ) determine the kernel of P:  ker P = f ∈ k[x] : λ j ( f ) = 0 for all j . Conversely, the kernel of P determines the span of (λ j ) since  span λ j = (ker P)⊥ . When the projector P is ideal, its kernel is often given by the ideal basis, a finite set of polynomials that generate the ideal ker P and thus the projector P is define by its values on a finite subsets of polynomials. The purpose of this note is to present a convenient formula (9) for symbolic computation of the functionals (λ j ) from these values. To expand on this point, recall Theorem 1 ([3]). A linear operator P : k[x] → k[x] is an ideal projector if and only if P( f g) = P( f · P(g)) (3) for all f , g ∈ k[x]. In terms of the quotient algebra k[x]/ ker P, (3) says that [ f [g]] = [ f g] ∈ k[x]/J, for all f , g ∈ k[x]. Let G ⊂ k[x] stand for a finite-dimensional range of the projector P, and let g = (g1 , . . . , gN ) be a linear basis for G. We define the border for g to be ∂ g := {1, xi gk , i = 1, . . . , d, k = 1, . . . , N}\G. By the de Boor’s formula (3), the ideal projector is completely determined by its finitely many values [3]: {P f , f ∈ ∂ g}. (4) Equivalently, the polynomials { f − P f , f ∈ ∂ g} form an ideal basis, called the border basis (cf. [5], [9], [11]) for the ideal ker P . The formula (9) computes the functionals λ j from the finitely many values of the ideal projector (4).

On Symbolic Computation of Ideal Projectors and Inverse Systems

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2 Preliminaries 2.1 Multiplication operators Using the polynomials (4) one can define a sequence of multiplication operators on G: MP,g = (M1 , . . . , Md ) where Mi (g) = P (xi g) .

(5)

With the aid of (3) it is easy to see (cf. [3], [11]) that this is a sequence of pairwise commuting operators with the cyclic vector P1: {p(MP,g )(P1), p ∈ k[x]} = G. The sequence MP,g is similar (literally and figuratively) to the operators of multiplication by xi on k[x]/J. There is a partial converse to this statement ([2], [11], [8]): every cyclic sequence L = (L1 , . . . , Ld ) of commuting operators on a finite-dimensional subspace G ⊂ k[x] defines the (unique) ideal projector. The sequence of multiplication operators for this projector is similar to L.

2.2 Duality The space of all formal power series in x is denoted by k[[x]]. For λˆ ∈ k[[x]], we use λˆ (D) to denote the differential operator on k[x] obtained by formally replacing the indeterminants with the corresponding partial derivatives with respect to these indeterminants. Every λˆ ∈ k[[x]] defines a linear functional λ on k[x] by   λ ( f ) := λˆ (D) f (0) for every f ∈ k[x]. It is well-known (cf. [4] and [6] in its original form) that the map λˆ 7−→ λ defined by the display above is a linear isomorphism between k[[x]] and (k[x])0 . Thus, every functional λ is identified with the power series λˆ ∈ k[[x]] and, when there is no possibility for confusion, we will denote both by the same letter. For instance, the point evaluation functional (k[x])0 3 λ : λ ( f ) := f (z) is identified with the power series for the exponential function λ (x) = ex·z . For a set J ⊂ k[x], we define J ⊥ := {λ ∈ k[[x]] : λ ( f ) = 0 for every f ∈ J}.

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Boris Shekhtman

For every f ∈ k[x] we have (cf. [4],[6]) (Di λ ) ( f ) = λ (xi f ) .

(6)

That is, the operator Di is the adjoint to the operator of multiplication by independent variable on k[x]. A linear subspace Λ ⊂ k[[x]] is D-invariant if Di λ ∈ Λ for every λ ∈ Λ and every i = 1, . . . , d. The next theorem (cf. [4]) is an easy consequence of (6): Theorem 2 ([6]). A subspace J ⊂ k[x] is an ideal if and only if J ⊥ ⊂ k[[x]] is Dinvariant.

3 The main result N

Theorem 3. Let P : P f = ∑ λ j ( f )g j be a N-dimensional ideal projector on k[x]. j=1
˜ P,g = M˜ 1 , . . . , M˜ d be the matriLet g = (g1 , . . . , gN ) be a basis for ran P and let M ces representing the operators Mi defined by (5) in the basis g. Then  t

λ := (λ1 , . . . , λN ) = e

d

∑ xi M˜ i

i=1



λ (0).

(7)

Proof. First, we claim that Mit = Di |G . Indeed, for every g ∈ G = ran P and every λ ∈ ran P∗ = (ker P)⊥ we have λ (Mi g) = (Mit λ ) (g). On the other hand λ (Mi g) = λ (P (xi g)) = (P∗ λ ) (xi g) = λ (xi g) = (Di λ ) (g), where the second equality follows from λ ∈ (ker P∗ )⊥ = ran P∗ and the last from (6). This means that N

(Di λk ) =

(i)

∑ m j,k λ j ,

(8)

j=1

N

(i)

(i)

where m j,k is the j, k-th entry in the matrix M˜ it . By D-invariance, Di λk = ∑ a j,k λ j (i)

j=1 (i)

(i)

for some coefficients a j,k . Since (λ j ) is a basis in ran P∗ it follows that a j,k = m j,k . The display (8) means that, as a vector-valued function of x1 , λ is the solution of the initial value problem D1 u = M˜ 1 u, u(0) = λ (0, x2 , . . . , xd ) . Thus

On Symbolic Computation of Ideal Projectors and Inverse Systems

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˜

˜

λ = ex1 M1 u(0) = ex1 M1 λ (0, x2 , . . . , xd ) . ˜

Next, we observe that, as a vector-valued function of x2 , ex1 M1 λ (0, x2 , . . . , xd ) solves the initial value problem ˜ D2 u = M˜ 2 u, u(0) = λ (x1 , 0, x3 , . . . , xd ) = ex1 M1 λ (0, 0, x3 , . . . , xd ) .

Hence ˜

˜

˜

˜

λ = ex1 M1 ex2 M2 λ (0, 0, x3 , . . . , xd ) = ex1 M1 +x2 M2 λ (0, 0, x3 , . . . , xd ) . Repeating this process d − 2 more times we obtain (7). Corollary 1. Suppose that g1 = 1 ∈ ran G. Then  t

λ = (λ1 , . . . , λN ) = e

d

∑ xi M˜ i i=1



e1 ,

(9)

where e1 = (1, 0, . . . , 0)t ∈ kN . Proof. Observe that for every λ ∈ k[[x]] we have λ (D)1 = λ (0). Since, by (2), (λ j (D)1) (0) = λ j (g1 ) = δ j,1 it follows that λ j (0) = δ j,1 . Remark 1. The formula (9) can be interpreted algebraically in terms of an inverse systems for the quotient ring of a zero-dimensional ideal. For a zero-dimensional ideal J ⊂ k[x], we define multiplication operators Mˆ i on k[x]/J by Mˆ i [ f ] = [xi f ] and the matrices M˜ i as the matrices of of the operators Mˆ i in any linear basis for k[x]/J. Then the formula (8) gives a linear basis for the Macaulay inverse systems (cf. [6]) for k[x]/J. The formula can also be interpreted as follows: Any N-dimensional D-invariant subspace of k[[x]] has
a basis of the form (9) for some cyclic sequence of commuting matrices M˜
1 , . . . , M˜ d . Conversely, every cyclic sequence of commuting matrices M˜ 1 , . . . , M˜ d generate an N-dimensional D-invariant subspace of k[[x]] via (9).

4 A couple of examples Both example will  deal with the ideal projector onto the span of g = (1, x, y), hence ∂ g = x2 , xy, y2 . Example 1. Define Px2 = Pxy = Py2 = 0. Then the matrices of multiplication operators in the basis g are     000 000 M1 = 1 0 0 , M2 = 0 0 0 . 000 100

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Boris Shekhtman

It is easy to verify that these matrices commute, hence define an ideal projector P. Maple computations give     1 1
exM1 +yM2 0 = x 0 y and P f = f (0)1 + (Dx f ) (0)x + (Dy f ) (0)y is the Taylor projector onto first degree polynomials. Example 2. Define Px2 = y, Pxy = Py2 = 0. Then the matrices of multiplication operators in the basis g are     000 000 M1 = 1 0 0 , M2 = 0 0 0 . 010 100 Again, it is easy to verify that these matrices commute, hence define an ideal projector P. Maple computations give     1 1
exM1 +yM2 0 =  x  1 2 0 2x +y and the ideal projector is given by  P f = f (0)1 + (Dx f ) (0)x +

1 2 D + Dy 2 x

 f (0)y

is the ideal projector in question.

References 1. Garrett Birkhoff. The algebra of multivariate interpolation. In Constructive approaches to mathematical models (Proc. Conf. in honor of R. J. Duffin, Pittsburgh, Pa., 1978), pages 345– 363. Academic Press, New York, 1979. 2. C. de Boor and B. Shekhtman. On the pointwise limits of bivariate Lagrange projectors. Linear Algebra Appl., 429(1):311–325, 2008. 3. Carl de Boor. Ideal interpolation. In Approximation theory XI: Gatlinburg 2004, Mod. Methods Math., pages 59–91. Nashboro Press, Brentwood, TN, 2005. 4. Carl de Boor and Amos Ron. On polynomial ideals of finite codimension with applications to box spline theory. J. Math. Anal. Appl., 158(1):168–193, 1991. 5. Martin Kreuzer and Lorenzo Robbiano. Computational commutative algebra. 2. SpringerVerlag, Berlin, 2005. 6. F. S. Macaulay. The algebraic theory of modular systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original, With an introduction by Paul Roberts.

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7. Hans Michael M¨oller. Hermite interpolation in several variables using ideal-theoretic methods. In Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), volume 571 of Lecture Notes in Math., pages 155–163. Springer, Berlin, 1977. 8. Hiraku Nakajima. Lectures on Hilbert schemes of points on surfaces, volume 18 of University Lecture Series. American Mathematical Society, Providence, RI, 1999. 9. L. Robbiano. On border basis and Gr¨obner basis schemes. Collect. Math., 60(1):11–25, 2009. 10. Tomas Sauer. Polynomial interpolation in several variables: lattices, differences, and ideals. In Topics in multivariate approximation and interpolation, volume 12 of Stud. Comput. Math., pages 191–230. Elsevier B. V., Amsterdam, 2006. 11. B. Shekhtman. Some tidbits on ideal projectors, commuting matrices and their applications. Elec. Trans. Numer. Anal., 36:17–26, 2009.