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The proper interpolation space for multivariate Birkhoff interpolation✩ Junjie Chai a,b , Na Lei a,b,∗ , Ying Li b , Peng Xia a a

School of Mathematics, Jilin University, Changchun 130012, PR China

b

Key Lab. of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University, Changchun 130012, PR China

article

info

Article history: Received 30 August 2010 Received in revised form 8 January 2011 Keywords: Birkhoff interpolation s-FLIT (the first s-linear independent terms) Minimal interpolation space Lower interpolation space

abstract Multivariate Birkhoff interpolation problem has many important applications, such as in finite element method. In this paper two algorithms are given to compute the basis of the minimal interpolation space and the lower interpolation space respectively for an arbitrary given node set and the corresponding interpolation conditions on each node. We can get the monomial basis, Newton-type basis as well as Lagrange-type basis. The interpolation polynomial can be derived from the basis directly. © 2011 Elsevier B.V. All rights reserved.

1. Introduction A multivariate Birkhoff interpolation scheme (Z , E , PS ) consists of three components. (a) A set of nodes Z , m Z = {zi }m i=1 = {(zi,1 , . . . , zi,n )}i=1 .

(b) An interpolation space PS ,

PS =

P | P (x) = P (x1 , . . . , xn ) =

−

α1

αn

aα x1 · · · xn

α=(α1 ,...,αn )∈S

where S is a subset of Nn0 , N = ♯S. (c) An incidence matrix Em×N E = (ei,α ),

i = 1 , . . . , m; α ∈ S ,

where ei,α = 0 or 1. The multivariate Birkhoff interpolation problem is to find a polynomial P ∈ PS satisfying

Li,α (P ) =

∂ α1 +···+αn P (zi ) = ci,α , α ∂ x1 1 · · · ∂ xαn n

∀ei,α = 1,

where ci,α are given constants, Li,α are functionals related to the corresponding interpolation conditions. The problem we consider is always normal, namely |E | = dim PS , where |E | =

∑m ∑N i=1

j =1

eij is the number of 1’s in E.

✩ This research was partly supported by the National Natural Foundation of China (No. 11001106).

∗

Corresponding author at: School of Mathematics, Jilin University, Changchun 130012, PR China. Tel.: +86 135 0443 1400. E-mail addresses: [email protected], [email protected] (N. Lei).

0377-0427/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2011.01.006

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If there exists a unique polynomial P ∈ PS satisfying the Birkhoff interpolation conditions, then we say the space PS is proper. The problem we focus on is: for a given node set Z and an incidence matrix E how to choose the proper interpolation space PS so that the interpolation polynomial always exists in it for any given constants ci,α . Turan proposed 15 questions on univariate Hermite–Birkhoff Interpolation problems in [1,2]. Shi solved some of them and presented his results in [3]. A great deal of research about univariate case focusses on specifical nodes or uniform interpolation conditions, such as [4–11]. Gonzalez-Vega [12] applied quantifier elimination to the Birkhoff interpolation problem. Palacios and Rubió [13] generalized the Pólya condition. Rubió studied the solvability of the univariate Birkhoff interpolation, gave out the interpolation space of the lowest degree [14], and also discussed the rank of univariate Birkhoff interpolation in [15]. Palacios et al. [16] discussed the order regularity of two-node univariate Birkhoff interpolation. Gasca and Martinez [17] used a matrix approach and the guideline of a result due to Stenger to solve the bivariate Hermite–Birkhoff interpolation problems. Carnicer and Gasca [18] gave a Newton-type interpolation formula for the enlarged problem. Crainic defined UR Birkhoff interpolation, that is a problem with rectangular sets of nodes and uniform interpolation conditions, and obtained quite a lot of results about it [19–26]. Lorentz [27] gave a general multivariate Birkhoff interpolation scheme and presented some results on the multivariate Birkhoff interpolation, including the necessary and sufficient condition for regular situation and the sufficient condition for almost regular situation about multivariate Birkhoff interpolation scheme. Wang et al. [28] obtained the Newton basis for an arbitrary x–y–z connected interpolation problem w.r.t. Hermite system. Marinari et al. [29] introduced an algorithm to compute the Groebner basis of a zero-dimensional ideal defined by a dual basis. This algorithm can also be used to compute the interpolation basis of Lagrange and Hermite interpolation problems, for the functionals determined by the homogeneous interpolation conditions define an ideal. Consequently Lagrange interpolation and Hermite interpolation are also called ideal interpolation (see [30]). But for the Birkhoff interpolation discussed in this paper, the algebraic structure defined by the functionals determined by the homogeneous interpolation conditions is not an ideal any more. It is just a subspace of polynomial ring. So the existing theory is totally unusable in Birkhoff interpolation problems. In this paper we generalize the theory in [29] to solve Birkhoff interpolation problems. The structure of this paper is as follows. We first introduce some definitions and theorems about K -vector space defined by functionals in Section 2, then in Section 3 we prove some main results. In Section 4 two algorithms are given to compute the minimal interpolation space and the lower interpolation space w.r.t. any given monomial order respectively. The algorithms present both monomial basis and Newton-type basis. The Lagrange-type basis and the interpolation polynomial can also be obtained easily. We conclude our results in the last section. 2. Preliminaries Let K be a field and K [X] be a polynomial ring. Considering K [X] as a K -vector space, a K -functional L over K [X] is a linear morphism L : K [X] → K , i.e. an element of K -vector space K [X]ˆ = HomK (K [X], K ). Definition 1 ([29]). Given s linear functionals L1 , . . . , Ls . We say a K -vector space Q ⊂ K [X] is defined by these functionals if Q = {f ∈ K [X] : Li (f ) = 0, 1 ≤ i ≤ s}. If {L1 , . . . , Ls } are linearly independent, then they are called a dual basis of Q . Let Q ⊂ K [X], then K [X]/Q be a quotient space. We say f and g are congruent modulo Q , if there exists a polynomial h ∈ Q , s.t. f = g + h. For any f ∈ K [X], [f ] which denotes the congruent class of f is the set followed

[f ] = {g ∈ K [X] : ∃h ∈ Q , s.t. f = g + h}. Definition 2. We say the congruent class [f1 ], [f2 ], . . . , [fs ] are linear dependent if there exist ci ∈ K (1 ≤ i ≤ s) which are not all zero such that c1 [f1 ] + · · · + cs [fs ] =

s − i=1

namely

∑s

i =1

ci [fi ] =

s − [ci fi ] = [0], i=1

ci fi ∈ Q .

Definition 3. We define Li ([f ]) , Li (f ), 1 ≤ i ≤ s. The above-mentioned definition is well defined. In fact given any f1 , f2 ∈ [f ], then f1 = f2 + h, where h ∈ Q . Hence, Li (f1 ) = Li (f2 + h) = Li (f2 ) + Li (h) = Li (f2 ), ∀i.

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Theorem 1 ([29]). Let Q be defined by linear functionals L1 , . . . , Ls , then dimK K [X]/Q ≤ s and equality holds if and only if {L1 , . . . , Ls } is a dual basis of Q . Definition 4 ([29]). If L1 , . . . , Ls ∈ sequence s.t. Li (qj ) =

0, 1,

K [X]ˆ, a biorthogonal sequence (b.o.s.) (q1 , . . . , qs ) of polynomials for L1 , . . . , Ls is a

i ̸= j, i = j.

Definition 5 ([29]). If L1 , . . . , Ls ∈ K [X]ˆ, a triangular sequence (t.s.) (q1 , . . . , qs ) of polynomials for L1 , . . . , Ls is a sequence s.t. Li (qj ) =

0, 1,

i < j, i = j.

Theorem 2 ([31]). Given L1 , . . . , Ls ∈ K [X]ˆ, a t.s. (q1 , . . . , qs ) of polynomials for L1 , . . . , Ls exists if and only if L1 , . . . , Ls are linearly independent. 3. Main results The Birkhoff interpolation problem is much more complicated than the Hermite interpolation problem [32] because the homogeneous interpolation space in this case is not an ideal any more. For a given node set Z and an incidence matrix E, our task is to find the proper space PS such that for any given constants ci,α , there always exists a unique polynomial in PS satisfying the interpolation conditions, and furthermore to get the expression of the interpolation polynomial. Definition 6. Let linear space Q ⊂ K [X], dim(K [X]/Q ) = s ≥ r. If a monomial sequence X = (X1 , . . . , Xr ) satisfies (i) (ii) (iii) (iv)

X1 ≺ · · · ≺ Xr w.r.t. given monomial order ≺; [X1 ], . . . , [Xr ] are linearly independent; ∀Z ≺ X1 , then [Z ] = [0]; ∑ if X1 ≼ Z ≺ Xr , then ∃ ai ∈ K , s.t. [Z ] = i ai [Xi ], {i : Xi ≼ Z , Xi ∈ X },

then we say X1 , . . . , Xr are the first r-linearly independent terms (r-FLIT for short) corresponding to Q . Theorem 3. Let linear space Q ⊂ K [X], dim(K [X]/Q ) = s ≥ r. r linearly independent monomials X1 ≺ · · · ≺ Xr are the first r-linearly independent terms corresponding to Q if and only if for any r linearly independent monomials Y1 , . . . , Yr satisfying Y1 ≺ · · · ≺ Yr , Xi ≼ Yi (1 ≤ i ≤ r ) hold. Proof. We prove the necessity by induction. If n = 1, we assume that Y1 ≺ X1 . According to (iii) of Definition 6, we have [Y1 ] = [0]. It follows that [Y1 ] is linear dependent. So [X1 ] ≼ [Y1 ]. Now we suppose the theorem is correct for n = r − 1. Let n = r, and X1 , . . . , Xr are the first r-linear independent terms. It is easy to see that X1 , . . . , Xr −1 are the first (r − 1)-linearly independent terms. With the independence of Y1 , . . . , Yr −1 we have Xi ≼ Yi , 1 ≤ i ≤ r − 1 by the assumption. Next we will show that Xr ≼ Yr . We assume that Yr ≺ Xr , then X1 ≼ Y1 ≺ Y2 ≺ · · · ≺ Yr ≺ Xr . By (iv) of Definition 6, we have

[Yj ] =

− i|Xi ≼Yj

(j)

(j)

(j)

ai [ X i ] =

r −1 −

(j)

bi [Xi ],

1 ≤ j ≤ r,

i=1

(j)

(j)

(j)

where bi = ai for Xi ≼ Yj and bi = 0 for Xi ≻ Yj . bi are not all zero. In fact, if bi are all zero, 1 ≤ i ≤ r − 1, then [Yj ] = [0]. This contradicts the linear independence of [Y1 ], . . . , [Yr ]. Thus [Y1 ], . . . , [Yr ] can be represented by [X1 ], . . . , [Xr −1 ]. This implies that [Y1 ], . . . , [Yr ] are linear dependent. It is a contradiction. Hence we have Xr ≼ Yr . Now we prove the sufficiency. (i) We suppose Y ≺ X1 , [Y ] ̸= [0]. Since dim(K [X]/Q ) = s ≥ r, we could find r − 1 monomials Y1 , . . . , Yr −1 such that [Y ], [Y1 ], . . . , [Yr −1 ] are linear independent. Permute Y , Y1 , . . . , Yr −1 w.r.t. ≺, we obtain Y1′ ≺ · · · ≺ Yr′ and [Y1′ ] . . . , [Yr′ ] are linear independent. But Y1′ ≼ Y ≺ X1 contradicts the condition. Hence [Y ] = [0]. (ii) If there exists a monomial Y satisfying Xi−1 ≺ Y ≺ Xi , 2 ≤ i ≤ r and cannot be represented by [X1 ], . . . , [Xi−1 ], then [Y ], [X1 ], . . . , [Xi−1 ] are linear independent. We can find r − i monomials Yi+1 , . . . , Yr , such that [X1 ], . . . , [Xi−1 ], [Y ], [Yi+1 ], . . . , [Yr ] are linear independent. Permute them according to ′ ≺′ we have Y1′ ≺ · · · ≺ Yr′ and Yi′ ≼ Y ≺ Xi , which contradicts the condition. This completes the proof.

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Theorem 4. Let dim(K [X]/Q ) = s. If T = {Ti , 1 ≤ i ≤ s} is s-FLIT corresponding to Q w.r.t. given monomial order ≺, then K [X]/Q ∼ = Span{T1 , . . . , Ts }. Proof. Define linear map f : Span{T1 , . . . , Ts } → K [X]/Q by f (t ) = [t ],

t ∈ Span{T1 , . . . , Ts }.

Since dim(K∑ [X]/Q ) = s and [T1 ], . . . ,∑ [Ts ] are linearly independent, they construct a basis of K [X]/Q . s s Let t = c T . If f ( t ) = [ 0 ] , i.e. i i i =1 i=1 ci [Ti ] = [0], then ci = 0, 1 ≤ i ≤ s. Hence t = 0. f is an injective. For any [u] ∈ K [X]/Q , ∃ ai , 1 ≤ i ≤ s, s.t. [u] = a1 [T1 ] + · · · + as [Ts ] = [a1 T1 + · · · + as Ts ], Obviously a1 T1 +· · ·+ as Ts ∈ Span{T1 , . . . , Ts }, and f (a1 T1 +· · ·+ as Ts ) = [u]. This implies that f is surjective. The proof is complete. Definition 7. We call a space PS the minimal interpolation space of a Birkhoff interpolation problem if

• PS is proper for a given node set Z and the incidence matrix E; • any of the monomials spanning PS cannot be minor w.r.t. the given monomial order. The monomials which span PS are called the minimal interpolation monomial basis. Theorem 5. Given a set of nodes Z and an incidence matrix Em×N , where |E | = s. ∀f ∈ K [X], define Li,α (f ) =

∂ α1 +···+αn f (zi ), α ∂ x1 1 · · · ∂ xαn n

for ei,α = 1.

Then the s functionals L1 , . . . , Ls define a K -vector space Q. Here the functionals are linearly independent as they are actually Birkhoff interpolation conditions. As a result dim(K [X]/Q ) = s. Let T1 , . . . , Ts are s-FLIT corresponding to Q w.r.t. given monomial order ≺. Then T = {T1 , . . . , Ts } is the minimal interpolation monomial basis of the Birkhoff interpolation problem. The space spanned by {T1 , . . . , Ts } is the minimal interpolation space. 4. Algorithms Now we present an algorithm to compute the minimal monomial basis w.r.t. the given monomial order ≺ and output a triangular sequence (t.s.) q1 , . . . , qs for a permutation L′1 , . . . , L′s of the original functionals. Note Definition 6, the t.s. is actually the Newton-type basis of the interpolation problem. In the algorithm below list X is iteratively updated which contains terms of s-FLIT, until the amount of the elements in X is s. The vector vect(t ) is associated consisting of the evaluations of monomial t at the functionals. At each step the minimal term t which is not in X is treated. If vect(t ) is linearly independent on {vect(m) : m ∈ X } then a new element is added to X , also a new element in a triangular sequence is obtained by Gaussian reduction. Algorithm 1. Input: L := L1 , . . . , Ls . Output: X := X1 , . . . , Xs and T := q1 , . . . , qs , where L1 , . . . , Ls are functionals defining the vector space Q ; X1 , . . . , Xs are s-FLIT of Q ; q1 , . . . , qs is a triangular sequence for a permutation L′1 , . . . , L′s of the original functionals. Start: L := L1 , . . . , Ls List := {1} (List is a term sequence ordered by the given monomial order ≺) X := ∅ (X is a list of r monomial terms X1 , . . . , Xr in increasing order) T := ∅ (T is a triangular sequence contains q1 , . . . , qr for L′1 , . . . , L′s , s.t. LT(qi ) = Xi and qi ∈ SpanK (X1 , . . . , Xi )) vect(i) = (L1 (qi ), . . . , Ls (qi )) While r ≤ s do t := Min(List, ≺) (t is the smallest monomial w.r.t. ≺ ) List := List-{t} v := (L1 (t ), . . . , Ls (t )) (q, v ) := Gauss-reduce(t , v ; q1 , . . . , qr ) If v ̸= 0, then r := r + 1 j := min(i : Li (q) ̸= 0)

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L′r := Lj vect(r ) := Lj (q)−1 v qr := Lj (q)−1 q X := X ∪ {t } Endif List := List ∪ {xj t , ∀j} Enddo End Theorem 6. Algorithm 1 eventually terminates. The output X = {X1 , . . . , Xs } and T = {q1 , . . . , qs } are the minimal interpolation monomial basis and the triangular sequence of the interpolation problem respectively. Proof. According to Theorem 5 there exist s independent monomials, so the algorithm terminates in s steps. Then we only need to prove that X = {X1 , . . . , Xs } obtained by Algorithm 1 are s-FLIT of Q w.r.t. monomial order ≺. From the algorithm we notice that the term t added into X each time is according to order ≺. Also by means of Gaussian reduction, [t ] must be linear independent on terms existing in the List. Hence X1 , . . . , Xs are linear independent. ∀Z , X1 < Z < Xs and Z ̸= Xi (1 ≤ i ≤ s), [Z ] must be linearly dependent on [Xi ](Xi ≤ Z , Xi ∈ X ), otherwise Z must be in X . By the definition of s-FLIT, X = {X1 , . . . , Xs } is the minimal monomial basis. Since the Gaussian matrix is an upper triangular matrix, T = {q1 , . . . , qs } is a triangular sequence. Hence we proved the correctness of the algorithm. Example 1. Let Z = {(1, 1), (1, 2), (2, 6)}, and

E=

1 1 0

1 1 0

0 1 0

0 0 1

0 0 0

0 0 . 1

By Algorithm 1 we compute that the s-FLIT are X = {1, y, x, y2 , x2 , y3 , xy2 } w.r.t. graded lex order, and the triangular sequence is 1; y − 1; x − 1; y2 − 2y + 1; 1 1 2 x −x+ ; y3 − 4y2 + 5y − 2; 2 2 1 2 1 xy − y2 − 2x + 2. 2 2 Then we easily obtain b.o.s. of the interpolation problem

−27xy2 + 2y3 + 18y2 + 108x + 12y − 112; −13xy2 + y3 + 8y2 + 52x + 8y − 56; 27xy2 − 2y3 − 18y2 − 108x − 12y + 113; −14xy2 + y3 + 10y2 + 56x + 5y − 58; x − 1; 1 2 1

xy2 −

1 2

y2 − 2x + 2;

1 x2 − x + . 2 2 Note the definition of the b.o.s., it is actually the Lagrange-type interpolation basis of the Birkhoff problem. (And the t.s. is actually the Newton-type basis.) If seven interpolation values {1, −2, 3, 9, −3, 0, −1} are given, then the interpolation polynomial can be written as f (x, y) = 1 × (−27xy2 + 2y3 + 18y2 + 108x + 12y − 112) + (−2) × (−13xy2 + y3 + 8y2 + 52x + 8y − 56)

+ 3 × (27xy2 − 2y3 − 18y2 − 108x − 12y + 113) + 9 × (−14xy2 + y3 + 10y2 + 56x + 5y − 58) 1 2 1 2 1 2 1 +(−3) × (x − 1) + 0 × xy − y − 2x + 2 + (−1) × x −x+ 2

1

2

= −46xy2 + 3y3 − x2 + 38y2 + 182x + 5y −

2

361

.

2

2 2 The minimal interpolation monomial basis we get by Algorithm 1 was shown in the left figure of Fig. 1. It is easy to prove that another monomial set Y = {1, y, x, y2 , x2 , y3 , y4 } (shown in the right figure of Fig. 1) is a proper interpolation basis, too. That is to say, the polynomial subspace spanned by Y = {1, y, x, y2 , x2 , y3 , y4 } is also proper for a given node set Z and the incidence matrix E. The set Y has a particular property, the indices of the monomials in Y construct a lower set.

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y

y

x

x

Fig. 1. Interpolation monomial basis.

Definition 8. A subset A of Nd0 is called lower if α = (α1 , . . . , αd ) ∈ A implies that R(α) = {β = (β1 , . . . , βd ) ∈ Nd0 : 0 ≤ βi ≤ αi , i = 1, . . . , d} ⊂ A. There are many fine properties of a lower monomial set (means that the set of the indices of the monomials is lower), such as the polynomial subspace spanned by it is closed under a differential operator, which is necessary in many practical cases. Next we study for a given node set Z and the incidence matrix E, whether there exists a lower interpolation space, and derive the algorithm to compute it in case of existing. Suppose S is a lower set of Nn0 , then if Y ∈ PS we have Z ∈ PS , ∀Z |Y . This means if Y ̸∈ PS , then W ̸∈ PS , ∀W : Y |W . So we may revise Algorithm 1 to get a lower set S ′ ∈ Nn0 . If |S ′ | = |E | then the space PS ′ is a proper interpolation space for the given Birkhoff problem. If |S ′ | < |E | then any PS with a lower set S ⊃ S ′ is not proper for Z and E. Algorithm 2. Input: L := L1 , . . . , Ls . Output: Y := Y1 , . . . , Yw and T := q1 , . . . , qw , w ≤ s, where L1 , . . . , Ls are s functionals defining the vector space Q ; Y1 , . . . , Yw are lower sets for the given functionals; q1 , . . . , qw is a triangular sequence for a permutation L′1 , . . . , L′w of the original functionals. Start: L := L1 , . . . , Ls List := {1} (List is a term sequence ordered by the given monomial order ≺) Y := ∅ (Y is a list of r monomial terms Y1 , . . . , Yr in increasing order) T := ∅ (T is a triangular sequence contains q1 , . . . , qr for L′1 , . . . , L′s , s.t. LT(qi ) = Yi and qi ∈ SpanK (Y1 , . . . , Yi ) ) v ect (i) = (L1 (qi ), . . . , Ls (qi )) While r ≤ s and List ̸= ∅ do t := Min(List, ≺) (t is the smallest monomial w.r.t. ≺) List := List-{t} If ∀i, t /xi ∈ Y v := L1 (t ), . . . , Ls (t ) (q, v ) := Gauss-reduce(t, v ; q1 , . . . , qr ) If v ̸= 0, then r := r + 1 j := min(i : Li (q) ̸= 0) L′r := Lj vect(r ) := Lj (q−1 )v qr := Lj (q−1 )q Y := Y ∪ {t } List := List ∪ {xj t , ∀j} Endif Endif Enddo End

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Theorem 7. Algorithm 2 eventually terminates. If w = s then Y = {Y1 , . . . , Yw } and T = {q1 , . . . , qw } are the lower monomial basis and the triangular sequence respectively. If w < s then any PS is not proper for Z and E with a lower set S ⊃ Y . Proof. The first part of the proof is similar to Algorithm 1. Let B = {Yi | Yi ∈ Y and ∃xh , s.t. Yi xh ̸∈ Y }. Then B is the border of the lower set. According to Algorithm 2 the set {Yi xj } ∪ Y is linearly dependent, where Yi ∈ B, 1 ≤ j ≤ n. Hence for any lower set S ⊃ Y , PS is not proper for given Z and E. We apply Algorithm 2 to compute the lower monomial basis. Example 2. We take the same node set and the incidence matrix as that in Example 1. Let Z = {(1, 1), (1, 2), (2, 6)}, and

E=

1 1 0

1 1 0

0 1 0

0 0 1

0 0 0

0 0 . 1

We obtain the lower monomial basis is X = {1, y, x, y2 , x2 , y3 , y4 }, which is indicated as the right figure in Fig. 1. The triangular sequence is 1;

y − 1; 1 2 1 y −y+ 2 2 1 3 1 2 2 5 y − y + y+ ; 27 6 9 54

x − 1; 1 2 1 x −x+ ; 2 2 27 4 404 3 1440 2 1776 713 y − y + y − y+ . 121 121 121 121 121 The b.o.s. of the interpolation problem is

1776 592 y2 + y− ; 121 121 121 774 2 1124 536 y4 + y3 − y + y− ; − 121 121 121 121 121 27 4 404 3 1440 2 1776 713 y − y + y − y+ ; 121 121 121 121 121 14 4 205 3 666 2 773 298 − y + y − y + y− ; 121 121 121 121 121 x − 1; 1 4 3 3 13 2 6 2 y − y + y − y+ ; 242 121 242 121 121 1 2 1 x −x+ . 2 2 If seven interpolation values {1, −2, 3, 9, −3, 0, −1} are given, then the interpolation polynomial is

−

27

121 13

y4 +

404 121 199

f ( x, y ) = 1 ×

1440

y3 −

−

27 121

404

y4 +

121

13

y3 −

1440 121

199

y2 + 774

1776 121

y−

592

121

1124

536

+ (−2) × − y + y − y + y− 121 121 121 121 121 27 4 404 3 1440 2 1776 713 +3 × y − y + y − y+ 121 121 121 121 121 14 4 205 3 666 2 773 298 +9 × − y + y − y + y− 121 121 121 121 121 1 4 3 3 13 2 6 2 1 2 1 + (−3) × (x − 1) + 0 × y − y + y − y+ + (−1) × x −x+ 4

3

242

=−

46 121

4

y +

639 121

3

y −

1 2

x −

Example 3. Let Z = {(1, 2), (3, 4)}, and

E=

1 0

0 1

0 0

0 0

0 0

1 . 1

2

1566 121

2

121

2

y − 2x +

242

1157 121

y+

121

479 242

.

121

2

2

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Y

1

0

1

2

X

Fig. 2. The first 4-linear independent terms.

From Algorithm 1 we get the first 4-linear independent terms are X = {1, y, x2 , x2 y} w.r.t. graded lex order (see Fig. 2). By Algorithm 2 we get Y = {1, y}. It is easy to see that for any PS satisfying dim PS = |E | = 4 and S is a lower set of Zn0 , PS is always not proper for Z and E. 5. Conclusion We introduced the first s-linear independent terms to construct the proper interpolation space for the multivariate Birkhoff interpolation problem. We present two algorithms to compute the minimal interpolation monomial basis and the lower interpolation monomial basis respectively. We can also get the Newton-type (t.s.) and Lagrange-type (b.o.s.) interpolation basis so that the interpolation polynomial can be derived directly. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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