Polynomial interpolation of minimal degree and ...

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Polynomial interpolation of minimal degree and Gröbner bases Thomas Sauer Mathematisches Institut, Universität Erlangen Bismarckstr. 1 12 , D–91054 Erlangen, Germany e-mail: [email protected] Abstract This paper investigates polynomial interpolation with respect to a finite set of appropriate linear functionals and the close relations to the Gröbner basis of the associated finite dimensional ideal.

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Introduction

In the 33 years since their introduction by Buchberger (1965, 1970), Gröbner bases have been applied successfully in various fields of Mathematics and to many types of problems. This paper wants to go the opposite way by presenting a different approach to Gröbner bases for zero dimensional ideals from the quite recent theory of polynomial interpolation of minimal degree. The latter one is an approach introduced by de Boor and Ron (1990, 1992) to solve interpolation problems defined by a finite number of linear functionals using appropriate polynomial spaces with certain useful properties. Let me briefly explain this with the example of Lagrange interpolation in Rd . Suppose that a finite set of pairwise disjoint points {x0 , . . . , xN } ∈ Rd is given. The Lagrange interpolation problem consists of finding, for any y0 , . . . , yN , a polynomial p such that p(xj ) = yj , j = 0, . . . , N . Clearly, this problem is always solvable and even has infinitely many solutions. The “real” question, however, is to find a polynomial subspace P such that for any given data the Lagrange interpolation problem has a unique solution in P and to choose P “as simple as possible”. For d = 1 the generic choice for P is obvious: one takes the polynomials of degree less than or equal to N . For d ≥ 2 the situation is more complicated since it may now happen that the points lie on an algebraic hypersurface of sufficiently low degree, for example when there are ≥ 6 points on a circle in R2 . In order to deal with this type of interpolation problem, one chooses the space P, depending on the data points, such that it satisfies additional minimality constraints. In the same sense as Gröbner bases associated to a term order are called standard bases for the ideal, one may ask for “standard” interpolation spaces

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associated to a certain term order. It will turn out that a natural choice of interpolation spaces provides a very close relationship to Gröbner bases; indeed, we will show that interpolating a polynomial from this interpolation space is equivalent to reducing the polynomial modulo the Gröbner basis.

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Polynomial interpolation of minimal degree

Let K be a field and denote by Π = K [ξ1 , . . . , ξd ] the ring of polynomials in d variables over K. A finite set Θ ⊂ Π0 of linear functionals mapping Π to K is said to admit an ideal interpolation scheme if the set IΘ := ker Θ = {p ∈ Π : Θ(p) = 0} ,

Θ(p) = (θ(p) : θ ∈ Θ) ,

is an ideal in Π. This terminology has been introduced by G. Birkhoff (1979). The interpolation problem associated to Θ then consists of finding a |Θ|– dimensional subspace P ⊂ Π and a projection L(P,Θ) : Π → P such that Θ L(P,Θ) q = Θ(q), q ∈ Π. (2.1) Note that if Θ is linearly independent this problem is equivalent to finding, for any y ∈ KΘ , a p = py ∈ P such that Θ(p) = y. Whenever (2.1) holds true, we call the pair (P, Θ) an interpolation system. The classical examples of ideal interpolation schemes are Lagrange interpolation (i.e., all functionals θj are the point evaluation functionals δxj for pairwise disjoint points in Rd ) or certain types of Hermite interpolation (i.e., interpolation of partial differential operators at pairwise disjoint points, cf. (Lorentz 1992) and (Sauer and Xu 1995), for example). As pointed out by Marinari et al. (1991) as well as de Boor and Ron (1992), ideal interpolation schemes can even be characterized as Hermite interpolation schemes with an additional closedness condition. The key notion for this result is that of a D–invariant polynomial subspace which means a subspace Q ⊂ Π with the property that p ∈ Π, q ∈ Q ⇒ p(D)q ∈ Q, where p(D) =

X

α∈Nd0

cα

∂ |α| , xα

p=

X

cα xα ∈ Π.

α∈Nd0

Theorem 1 Let Θ ⊂ Π0 be a finite set of linearly independent functionals. Then IΘ is an ideal if and only if there are points x1 , . . . , xm ∈ Kd and D– invariant subspaces Q1 , . . . , Qm ⊂ Π such that spanK Θ = spanK δxj ◦ qj (D) : qj ∈ Qj , j = 1, . . . , m .

Polynomial interpolation and Gröbner bases

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It is also known, cf. (de Boor and Ron 1991), that the “local” Hermite interpolation conditions connected to some point xj correspond to the primary decomposition of the ideal IΘ . Examining the structure of the spaces Qj , j = 1, . . . , m, leads to a refined notion of the multiplicity of a zero, cf. (Marinari et al. 1996). Let ≺ be any total order of the multiindices Nd0 which is compatible with addition and satisfies 0 α for any α ∈ Nd0 . Clearly, this well–ordering induces a term order on the monomials xα , α ∈ Nd0 , and therefore the notion of a leading term Λ(p) = Λ≺ (p) of a polynomial p ∈ Π defined by X Λ(p) = max {cα xα : cα 6= 0} , p= cα xα . ≺

α∈Nd0

Finally, we write p ≺ q for p, q ∈ Π if Λ(p) ≺ Λ(q) and interpret this as p being of lower degree than q. Let Θ define an ideal interpolation scheme. Then we call a subspace P ⊂ Π a minimal degree interpolation space for Θ with respect to the term order ≺ if 1. (P, Θ) is an interpolation system, i.e., for any y ∈ KΘ there is a unique p ∈ P such that Θ(p) = y, 2. P is ≺–minimal with this property, i.e., there exists no interpolation system (Q, Θ) with Q ≺ P. The latter means that for any q ∈ Q there exists a p ∈ P such that q ≺ p. 3. P is ≺–reducing, i.e., when L(P,Θ) denotes the projection on P (the interpolation polynomial), then L(P,Θ) q q,

q ∈ Π.

In general, there is no unique minimal interpolation space for a set of interpolation conditions Θ. More precisely, we have the following result. Proposition 2 Let Θ be a linearly independent set of linear functionals admitting an ideal interpolation scheme. The the following statements are equivalent: 1. There exists a unique minimal degree interpolation space P for Θ with respect to the term order ≺. 2. Let A|Θ| ⊂ Nd0 denote the first |Θ| multiindices with respect to the ordering ≺, then the Vandermonde matrix [θ ((·)α )]θ∈Θ, α∈A|Θ| has full rank.

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T. Sauer 3. P = spanK xα : α ∈ A|Θ| .

Proof: We first note that if spanK xα : α ∈ A|Θ| is an interpolation space, then it is clearly a minimal degree interpolation space, since n o dim spanK xα : α ≺ max A|Θ| = |Θ| − 1. ≺

Consequently, any other interpolation space must contain a polynomial p such that p xα , α ∈ A|Θ| , and therefore it cannot be minimal. Hence, conditions 2. and 3. imply uniqueness. Conversely, suppose that P is a minimal degree interpolation space, let p0 ≺ · · · ≺ pN be a basis of P and suppose that there exist a polynomial q ∈ Π \ P such that q ≺ pN . Since q 6∈ P and since the projection L(P,Θ) is degree reducing we conclude that 0 6= q − L(P,Θ) q q ≺ pN . But then, for any 0 6= c ∈ K, we have that spanK p0 , . . . , pN −1 , pN + c q − L(P,Θ) q 6= P is another minimal degree interpolation space. 2 The above result indicates that a “good” or “natural” choice for a interpolation space would be one consisting of exactly N + 1 monomials.

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Newton basis and Gr¨ obner basis

Our next goal is to derive a suitable basis for the minimal degree interpolation space and relate that to Gröbner bases. To simplify notation, we set N := |Θ| − 1 and write Θ = {θ0 , . . . , θN }, where we assume the indexing to be chosen in a proper way which will be specified later. We will also use the notation Θj = (θ0 , . . . , θj ), j = 0, . . . , N , for interpolation subproblems. Also for any polynomial p ∈ Π we define the linear subspace of polynomials of the same or lower degree as Πp = {q ∈ Π : q p}

and

Π≺p = {q ∈ Π : q ≺ p} .

Clearly, p ∈ Πp . A Newton basis of a polynomial subspace P with respect to Θ is a set of polynomials p0 ≺ · · · ≺ pN ∈ P such that, after numbering Θ properly, the following properties are satisfied: 1. P = spanK {p0 , . . . , pN }, 2. θi (pj ) = δij , 0 ≤ i ≤ j ≤ N ,


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3. there exist polynomials q0 , . . . , qM ∈ IΘ such that ΠpN = spanK {p0 , . . . , pN } ⊕ spanK {q0 , . . . , qM } .

(3.1)

Recalling the terminology of Marinari et al. (1991), the polynomials p0 , . . . , pN are a triangular sequence with respect to Θ. The additional condition 3. will correspond to minimality. The name Newton basis stems from the fact that the interpolation polynomial can now be written in the iterative “Newton form” L(P,Θ) q =

N X j=0

θj q − L(Pj−1 ,Θj−1 ) q pj ,

Pj = spanK {p0 , . . . , pj } .

Indeed we have the following connection between minimal degree interpolation spaces and Newton bases. Theorem 3 Let Θ be a finite linearly independent set of linear functionals admitting an ideal interpolation scheme. Then P is a minimal degree interpolation space with respect to Θ if and only if it has a Newton basis with respect to Θ. Proof: The proof is similar to the one for the total degree situation (Sauer 1997). Let us first assume that P has a Newton basis, then the interpolation property is immediate and from equation (3.1) we observe that rank [θ ((·)α )]θ∈Θ, xα ≺pN = rank [θ(q0 ), . . . , θ(qM ), θ(p0 ), . . . , θ(pN −1 )]θ∈Θ = rank [θ(p0 ), . . . , θ(pN −1 )]θ∈Θ = |Θ| − 1, yielding that P is ≺–minimal. Since Π = P ⊕ IΘ , we can write any p ∈ Π as p=

N X j=0

aj p j +

M X

bj qj + q,

q ∈ IΘ ,

j=0

and therefore we also conclude that L(P,Θ) p =

N X

aj pj p,

j=0

hence P is also ≺–reducing and therefore it is indeed a minimal degree interpolation space for Θ. Conversely, if P is a minimal degree interpolation space for Θ, then P has dimension |Θ| = N + 1 and therefore there exists a “graded” basis p˜0 ≺ · · · ≺

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pÑ of P. Gauß elimination on the Vandermonde matrix then immediately yields that, after numbering Θ properly, there exist a triangular basis p0 ≺ · · · ≺ pN for P. Now, let q˜0 ≺ · · · ≺ q˜M be linearly independent polynomials which complete p0 , . . . , pN to a basis of ΠpN . Clearly, q˜j ≺ pN . Since P is ≺– reducing, we have for any j = 0, . . . , M that q˜j L(P,Θ) q˜j , hence q˜j qj q˜j , where qj := q˜j − L(P,Θ) q˜j ∈ IΘ , j = 0, . . . , M, while still ΠpN = spanK {p0 , . . . , pN } ⊕ spanK {q0 , . . . , qM } . This verifies that p0 , . . . , pN is a Newton basis. 2 Let us now recall that a basis of a polynomial ideal I, i.e., a finite set of polynomials f1 , . . . , fn such that ( n ) X I = hf1 , . . . , fn i = qj fj : qj ∈ Π, j = 1, . . . , n , j=1

is called a Gr¨ obner basis of I if hΛ(f1 ), . . . , Λ(fn )i = hΛ(I)i := {Λ(q) : q ∈ I} . It is well–known, cf. (Buchberger and Möller 1982), (Marinari et al. 1991), as well as (de Boor 1994), that a triangular system for Θ and a Gröbner basis for the ideal IΘ can be constructed simultaneously by properly applying Gauß elimination to the Vandermonde matrix. An equivalent approach is to use a Gram–Schmidt orthogonalization process for the monomials and successively factor out the subideals generated by the Gröbner basis elements obtained during this process; this has been described explicitly for the case of the graded lexicographical ordering (Sauer 1997) Let me briefly outline how the latter approach works for finding the Newton basis (and therefore the “natural” interpolation space PΘ announced earlier in this paper) and the reduced Gröbner basis for IΘ simultaneously. For that purpose we first remark that Θ defining an ideal interpolation scheme yields that any minimal degree interpolation space P is a subspace of ΠN , the space of all polynomials of total degree at most N and that therefore all elements of the reduced Gröbner basis can be found in ΠN +1 . This elementary remark gives us an a priori bound on the total degree of the Gröbner and Newton basis elements which enables us to formulate the algorithm in a “finite environment”. We first initialize the polynomials φα = xα , |α| ≤ N + 1, let Θ0 = Θ be the set of all “free” linear functionals from Θ and set j = 0. Now we loop over all multiindices α of absolute value ≤ N +1 according to the ordering ≺ and check if Θ0 (φα ) = 0. If this is the case and φα 6= 0, then we mark φα as a member of


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the Gröbner basis and set φα+β = 0, β ∈ Nd0 , |α + β| ≤ N + 1. Otherwise, we pick θ ∈ Θ0 (which leaves room for a pivoting strategy) such that θ(φα ) 6= 0, set θj = θ, pj = φα /θj (φα ), do the orthogonalization φβ ← φβ − θj (φβ )pj , β α, |β| ≤ N + 1 and continue with Θ0 ← Θ0 \ {θ} and j ← j + 1. It is easy to see that the interpolation space generated this way is spanned by exactly |Θ| monomials and that the Gröbner basis elements are of the form xα + p, xα p ∈ P. In particular, no leading term of a Gröbner basis element divides any term of another one which verifies the claim that the Gröbner basis generated by this algorithm is the reduced Gröbner basis. Conversely, let GΘ = {g1 , . . . , gn } denote the unique reduced Gröbner basis for IΘ , i.e., the leading terms Λ(gj ), j = 1, . . . , n, have coefficient 1 and no leading term of one of these polynomials divides any term of another one. Since the minimal degree interpolation space generated by the above construction satisfies P = Π Λ(IΘ ) = Π hΛ(GΘ )i ,

(3.2)

there are exactly N + 1 monomials not contained in Λ(IΘ ). Hence we have the following result. Theorem 4 Let Θ be a finite linearly independent set of linear functionals admitting an ideal interpolation scheme. Then there exists a unique minimal degree interpolation space PΘ which is spanned by |Θ| monomials. Clearly, the ideal IΘ can also be written as IΘ = p − L(PΘ ,Θ) p : p ∈ Π = xα − L(PΘ ,Θ) (·)α : α ∈ N0 . α d Set qα = xα − L (PΘ ,Θ) (·) , α ∈ N0 . Since IΘ dis an ideal, the set AΘ := d α ∈ N0 : qα 6= 0 is an upper set, i.e., AΘ + N0 ⊂ AΘ . Then it is easily observed that the corners CΘ ⊂ AΘ , i.e., those elements α ∈ AΘ which cannot be written as α = α0 + β, α0 ∈ AΘ , 0 6= β ∈ Nd0 , point to the elements of the Gröbner basis. In other words, GΘ = xα − L(PΘ ,Θ) (·)α : α ∈ CΘ . (3.3)

A careful investigation of these relationships has been performed by Marinari et al. (1991). Let us briefly comment on some algorithmic aspects of the above algorithm: of course, “sifting” the full ΠN +1 in order to obtain the Newton basis and the Gröbner basis, may usually cause a lot of unnecessary effort. A more effective approach will proceed by degree and if, for some k ∈ N, the space Πk is not sufficient for the Newton basis (in terms of the above procedure this means that Θ0 6= ∅), then one might add the polynomials φα = xα − LPk ,Θ\Θ0 , |α| = k + 1, where Pk ⊂ Πk is the interpolation space spanned

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by the already constructed Newton polynomial which interpolates at Θ \ Θ0 . Moreover, in the above “addition technique” those monomials can be omitted which are divisible by the leading term on an already found element of the Gröbner basis. Information on how to practically compute interpolation polynomials by means of the Newton method, including a triangular scheme for the coefficients of the Newton representation, can be found in (Sauer 1995).

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Interpolation and reduction

Gröbner bases are closely connected to (and most frequently even defined by) the notion of reduction. We say the a polynomial p ∈ Π reduces to q ∈ Π modulo a finite set F = {f1 , . . . , fn } ⊂ Π if there exist g1 , . . . , gn ∈ Π such that n X p= fj gj + q j=1

and none of Λ(f1 ), . . . , Λ(fn ) divides any term of q. It is well–known that if G is a Gröbner basis, then the remainder of reduction is unique and therefore reduction with respect to a Gröbner basis G can be understood as a mapping →G : Π → Π. Moreover, we call a polynomial p ∈ Π reduced with respect to a Gröbner basis G if p = p →G . Reduced polynomials can easily be described in the following way. Lemma 5 Let G be a Gr¨ obner basis for the ideal I. A polynomial p ∈ Π is reduced modulo G if and only if g ∈ I, Λ(g)(D)p = 0, (4.1) g ∈ G. Proof: Equation (4.1) with g ∈ G is only a reformulation of the assumption that no term of p is divisible by any Λ(g), g ∈ G. Also, the validity of (4.1) for all g ∈ I trivially implies the validity for g ∈ G. Conversely, since G is a Gröbner basis, there exist, for any q ∈ I, monic polynomials qg , g ∈ G, such that X Λ(q) = Λ(qg )Λ(g) g∈G

and therefore, for any reduced polynomial p ∈ Π X Λ(q)(D)p = Λ(qg )(D) (Λ(g)(D)p) = 0. {z } | g∈G

=0

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From this lemma it immediately follows that the reduced polynomials form a D–invariant finite dimensional subspace which has the form Π Λ(IΘ ). Recalling equation (3.2) we therefore obtain the representation \ \ ker Λ(q)(D) = ker Λ(g)(D), (4.2) PΘ = g∈GΘ

q∈IΘ

which implies the following result, cf. (Möller 1998). Theorem 6 Let Θ ⊂ Π0 be a finite linearly independent set which admits an ideal interpolation scheme. A polynomial p ∈ Π is reduced modulo GΘ if and only if p ∈ PΘ . Moreover, for any p ∈ Π we have L(PΘ ,Θ) p = p →GΘ .

(4.3)

Proof: The first statement follows from Lemma 5 and (4.2). To prove (4.3) we first note that on PΘ both mappings act as the identity while for general p ∈ Π we write X p = p →G + pg g g∈G

and then L(PΘ ,Θ) p = L(PΘ ,Θ) (p →G ) +

X

L(PΘ ,Θ) (pg g) = p →G ,

g∈G

since Θ(pg g) = 0, g ∈ G. 2 Consequently, by switching between interpolation and reduction it is possible by simple and efficient algorithms to compute either p →GΘ from the data Θ(p) or L(PΘ ,Θ) p if only GΘ (but not Θ itself) is known.

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Least Interpolation

There is a way to write the space PΘ in a “closed form” which has been used by de Boor and Ron to introduce their concept of least interpolation (de Boor and Ron 1990). Following their approach (de Boor and Ron 1992), we identify Π0 with the formal power series K[[ξ1 , . . . , ξd ]] and use the pairing h·, ·i : K[[ξ1 , . . . , ξd ]] × K[ξ1 , . . . , ξd ] → K defined by hf, pi =

X Dα f (0)Dα p(0) = p(D)f (0). α! d

α∈N0

We can identify any θ ∈ Θ with an fθ ∈ K[[ξ1 , . . . , ξd ]] (for example, if θ = δx , then fθ (y) = ex·y ). Then the following variant of Theorem 1 is valid; cf. (de Boor and Ron 1992), where also the proof is taken from which we give for the sake of completeness.

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Theorem 7 The finite set Θ ⊂ Π0 admits an ideal interpolation scheme if and only if the subspace fΘ := spanK {fθ : θ ∈ Θ} is closed under formal differentiation. Proof: The proof is based on the simple observation that, for α ∈ Nd0 , hf, (·)α pi = (p(D)Dα f ) (0) = hDα f, pi ,

(5.1)

which holds for any p ∈ K[ξ1 , . . . , ξd ], f ∈ K[[ξ1 , . . . , ξd ]]. Hence, if fΘ is closed under differentiation, then kerΘ is obviously an ideal by (5.1). Conversely, since fΘ is of finite dimension, it is closed and therefore fΘ = {f ∈ K[[ξ1 , . . . , ξd ]] : hf, pi = 0, p ∈ kerΘ} , we can again use (5.1) to conclude that fΘ is closed under differentiation. 2 For any f ∈ K[[ξ1 , . . . , ξd ]] we denote by λ(f ) = λ≺ (f ) the least term of f with respect to the term order ≺ as the ≺–minimal nonzero term of the power series f , i.e., X λ(f ) = min {fα xα : cα 6= 0} , f= fα xα . ≺

α∈Nd0

This notion gives the desired representation of PΘ . Theorem 8 Assume that the finite and linearly independent set Θ ⊂ Π0 admits an ideal interpolation scheme. Then PΘ = λ (fΘ ) = {λ(f ) : f ∈ fΘ } . Proof: We first P prove that λ(fΘ ) ⊂ PΘ . For that purpose we assume that there exist f = cθ fθ , cθ ∈ K, θ ∈ Θ, and q ∈ IΘ such that Λ(q)(D)λ(f ) 6= 0. This implies that q λ(f ) q and therefore X X 0 6= Λ(q)(D)λ(f ) = q(D)f (0) = hf, qi = cθ hfθ , qi = cθ θ(q) = 0, θ∈Θ

θ∈Θ

which is a contradiction. Hence, Λ(q)(D)λ(f ) = 0, q ∈ IΘ , f ∈ λ(Θ) and therefore \ λ(fΘ ) ⊂ ker Λ(q)(D) = PΘ . q∈IΘ

On the other hand, the functionals Θ were assumed to be linearly independent which implies that dim λ(fΘ ) = dim fΘ = dim Θ = |Θ| = dim PΘ , hence λ(fΘ ) = PΘ . 2


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Sauer, T. (1995) ‘Computational aspects of multivariate polynomial interpolation’, Advances Comput. Math. 3, 219–238. Sauer, T., Xu, Yuan (1995) ‘On multivariate Hermite interpolation’, Advances Comput. Math. 4, 207–259. Sauer, T. (1997) ‘Polynomial interpolation of minimal degree’, Numer. Math., to appear.