Groebner basis structure of ideal interpolation
aa r X i v : . [ c s . S C ] J u l GR ¨OBNER BASIS STRUCTURE OF IDEAL INTERPOLATION ∗ YIHE GONG † AND
XUE JIANG ‡ Abstract.
We study the relationship between certain Gr¨obner bases for zero dimensional ideals,and the interpolation condition functionals of ideal interpolation. Ideal interpolation is defined by alinear idempotent projector whose kernel is a polynomial ideal. In this paper, we propose the notionof “reverse” complete reduced basis. Based on the notion, we present a fast algorithm to computethe reduced Gr¨obner basis for the kernel of ideal projector under an arbitrary compatible ordering.As an application, we show that knowing the affine variety makes available information concerningthe reduced Gr¨obner basis.
Key words.
Gr¨obner basis, ideal interpolation, multivariate polynomial, monomial ordering
AMS subject classifications.
1. Introduction.
Let F be either the real field R or the complex field C . Poly-nomial interpolation is to construct a polynomial g belonging to a finite-dimensionalsubspace of F [ X ] from a set of data that agrees with a given function f at the dataset, where F [ X ] := F [ x , x , . . . , x d ] denotes the polynomial ring in d variables overthe field F .For studying multivariate polynomial interpolation, Birkhoff [1] introduced thedefinition of ideal interpolation. Ideal interpolation can be defined by a linear idem-potent projector whose kernel is a polynomial ideal. In ideal interpolation [7], theinterpolation condition functionals at an interpolation point θ ∈ F d can be describedby a linear space span { δ θ ◦ P ( D ) , P ∈ P θ } , where P θ is a D -invariant polynomialsubspace, δ θ is the evaluation functional at θ and P ( D ) is the differential operator in-duced by P . The classical examples of ideal interpolation are Lagrange interpolationand Hermite interpolation. As pointed out by de Boor [7] and Shekhtman [9], idealinterpolation provides a natural link between polynomial interpolation and algebraicgeometry.For an ideal interpolation, suppose that ∆ is the finite set of interpolation con-dition functionals, then the set of all polynomials that vanish at ∆ constitutes a zerodimensional ideal, denoted by I (∆). Namely, I (∆) := { f ∈ F [ X ] : σ ( f ) = 0 , ∀ σ ∈ ∆ } . Gr¨obner bases, introduced by Buchberger [2] in 1965, have been applied suc-cessfully in various field of mathematics and to many types of problems. As is wellknown, the most significant milestone of the computation of vanishing ideals is theBM algorithm [3]. For any point set Θ ⊂ F d and fixed monomial ordering ≺ [5], theBM algorithm yields the reduced Gr¨obner basis for I (Θ) w.r.t. ≺ . Another famousalgorithm is referred as MMM algorithm [4].Given a monomial ordering ≺ and ideal interpolation condition functionals ∆, wegive a method to find the reduced Gr¨obner basis for I (∆) w.r.t. ≺ . The paper is ∗ Submitted to the editors DATE.
Funding:
This work was supported by National Natural Science Foundation of China underGrant No. 11671169 and 11901402. † Corresponding author, College of Science, Northeast Electric Power University, Jilin, China([email protected]). ‡ School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, China([email protected]). 1
Y. H. GONG, X. JIANG organized as follows. The algorithm for “reverse” complete reduced basis in section 3.The method to find interpolation monomial basis and quotient ring basis in section 4.The algorithm for Gr¨obner basis’s leading monomials in section 5. Our main resultsthe algorithms for reduced Gr¨obner bases in section 6, and the conclusions follow insection 7.
2. Preliminaries.
Throughout the paper, Z ≥ denotes the set of nonnegativeintegers. Let Z d ≥ := { ( α , α , . . . , α d ) | α i ∈ Z ≥ } . For α := ( α , α , . . . , α d ) ∈ Z d ≥ , α ! := α ! α ! · · · α d ! and we write X α for the monomial x α x α · · · x α d d . A polynomial P ∈ F [ X ] can be considered as the formal power series P = X α ∈ Z d ≥ ˆ P ( α ) X α , where ˆ P ( α )’s are the coefficients in the polynomial P . P ( D ) := P ( D x , D x , . . . , D x d ) is the differential operator induced by the poly-nomial P , where D x j := ∂∂x j is the differentiation with respect to the j th variable, j = 1 , , . . . , d .Let D α := D α x D α x · · · D α d x d , the differential polynomial is defined as P ( D ) := X α ∈ Z d ≥ ˆ P ( α ) D α . Given a monomial ordering ≺ , the leading monomial of a polynomial P ∈ F [ X ]w.r.t. ≺ is defined by LM( P ) := max ≺ { X α | ˆ P ( α ) = 0 } , and the least monomial of the polynomial P w.r.t. ≺ is defined bylm( P ) := min ≺ { X α | ˆ P ( α ) = 0 } . For example, given the monomial ordering grlex( y ≺ x ), for P = x + xy + y ∈ F [ x, y ], we have LM( P ) = x , lm( P ) = y. Definition { P , P , . . . , P n } the set of all monomials thatoccur in the polynomials P , P , . . . , P n with nonzero coefficients.For example, let P = 1 , P = x, P = x + y , thenΛ { P , P , P } = { , x, y, x } . Definition ≺ , { P , P , . . . , P n } ⊂ F [ X ] is calleda complete reduced basis w.r.t. ≺ , if1. P , P , . . . , P n are linearly independent,2. LM( P i ) Λ { P j } , i = j, ≤ i, j ≤ n . Example Given the monomial ordering grlex( z ≺ y ≺ x ) , { , x, x + y, x + xy + y } , { , y + z, x } , { , y + z, x + z } are complete reduced bases w.r.t. ≺ . ROBNER BASIS Definition ≺ , { P , P , . . . , P n } ⊂ F [ X ] is calleda “reverse” complete reduced basis w.r.t. ≺ , if1. P , P , . . . , P n are linearly independent,2. lm( P i ) Λ { P j } , i = j, ≤ i, j ≤ n . Example Given the monomial ordering grlex( z ≺ y ≺ x ) , { , x, x + y, x − x + xy } , { , y + z, x } , { , x + z, x − y } are “reverse” complete reduced bases w.r.t. ≺ .
3. The algorithm to compute a “reverse” complete reduced basis.
Givena monomial ordering ≺ and linearly independent polynomials P , P , . . . , P n ∈ F [ X ],Algorithm 3.1 yields a “reverse” complete reduced basis w.r.t. ≺ . Algorithm 3.1
A “reverse” complete reduced basis w.r.t. ≺ Input:
A monomial ordering ≺ . Linearly independent polynomials P , P , . . . , P n ∈ F [ X ]. Output: { P ( n )1 , P ( n )2 , . . . , P ( n ) n } , a “reverse” complete reduced basis w.r.t. ≺ . //Initialization P (0)1 := P , P (0)2 := P , . . . , P (0) n := P n . //Computing for k = 1 : n do P ( k ) k = P ( k − k ; for j = 1 : n, j = k do X β ( k − k = lm( P ( k − k ); P ( k ) j = P ( k − j − (cid:18) ˆ P ( k − j ( β ( k − k )ˆ P ( k − k ( β ( k − k ) (cid:19) P ( k − k ; end for end for return { P ( n )1 , P ( n )2 , . . . , P ( n ) n } ;Notice that ˆ P ( k − k ( β ( k − k ) is the coefficient of the least monomial of P ( k − k , so itis nonzero. It is obvious that Algorithm 3.1 terminates. The following theorem showsits correctness. Theorem
Given a monomial ordering ≺ , a set of linearly independent poly-nomials can be transformed into a “reverse” complete reduced basis w.r.t. ≺ .Proof. Suppose that P , P , . . . , P n ∈ F [ X ] are linearly independent polynomials,we only need to prove that1. P ( n )1 , P ( n )2 , . . . , P ( n ) n are linearly independent,2. lm( P ( n ) k ) Λ { P ( n ) j } , k = j, ≤ k, j ≤ n .According to Line 8 and 11 in Algorithm 3.1, it is easy to check that { P ( t − j , ≤ j ≤ n } can be expressed linearly by { P ( t ) j , ≤ j ≤ n } , 1 ≤ t ≤ n, hence { P j , ≤ j ≤ n } can be expressed linearly by { P ( n ) j , ≤ j ≤ n } . Notice that { P j , ≤ j ≤ n } arelinearly independent, so { P ( n ) j , ≤ j ≤ n } are also linearly independent. Y. H. GONG, X. JIANG
According to Line 8 and 11 in Algorithm 3.1, it is obvious that(3.1) lm( P ( k ) k ) = lm( P ( k − k ) = X β ( k − k Λ { P ( k ) j } , k = j, ≤ k, j ≤ n, and it is easy to check that lm( P ( k ) k ) = lm( P ( k +1) k ) = · · · = lm( P ( n ) k ) , ≤ k ≤ n. According to Line 8 and 11 in Algorithm 3.1, we haveΛ { P ( t ) j , ≤ j ≤ n, j = k } ⊆ Λ { P ( k ) j , ≤ j ≤ n, j = k } , ≤ k ≤ t ≤ n. Thus by (3.1) we havelm( P ( n ) k ) = lm( P ( k ) k ) Λ { P ( n ) j , ≤ j ≤ n, j = k } ⊆ Λ { P ( k ) j , ≤ j ≤ n, j = k } , ≤ k ≤ n. i.e. lm( P ( n ) k ) Λ { P ( n ) j } , k = j, ≤ k, j ≤ n. We now see { P ( n )1 , P ( n )2 , . . . , P ( n ) n } is a “reverse” complete reduced basis w.r.t. ≺ . Example Given the monomial ordering grlex( y ≺ x ) . Let { P , P , P , P } = { , x, x +2 y, x + xy + y } ⊂ F [ x, y ] , they are linearly independent. By Algorithm ,we get { P , P , P , P − P } = { , x, x + 2 y, x − x + xy } is a “reverse” complete reduced basis w.r.t. ≺ .Remark
4. Interpolation monomial basis and quotient ring basis.
Definition
Let T and T ′ be two sets of monomials in F [ X ] with T ′ − T = ∅ and T − T ′ = ∅ . Given a monomial ordering ≺ , we call T ′ ≺ T , if max ≺ ( T ′ − T ) ≺ max ≺ ( T − T ′ ) . For example, let T = { , y, xy, x } , T ′ = { , x, y , x } , then T ′ − T = { x, y } , T − T ′ = { y, xy } . Given the monomial ordering grlex( y ≺ x ), we havemax ≺ ( T ′ − T ) = y ≺ xy = max ≺ ( T − T ′ ) , and it means T ′ ≺ T . Definition ≺ -minimal monomial basis [6]). Given a monomial ordering ≺ and interpolation conditions ∆ = δ ◦ { P ( D ) , P ( D ) , . . . , P n ( D ) } , let T be an inter-polation monomial basis for ∆ , then T is ≺ -minimal if there exists no interpolationmonomial basis T ′ for ∆ satisfying T ′ ≺ T . Lemma
Given interpolation conditions ∆ = δ ◦ { P ( D ) , P ( D ) , . . . , P n ( D ) } and a set of monomials T = { X β , X β , . . . , X β n } , the matrix applying T on ∆ isdenoted by T ∆ := ( δ ◦ P i ( D ) X β j ) ij , ≤ i, j ≤ n. Then T is an interpolation monomial basis for ∆ iff T ∆ is non-singular.Proof. Suppose that the interpolating polynomial g = P nj =1 c j X β j and the valuesare f i ’s, i = 1 , , . . . , n . It means( δ ◦ P i ( D )) g = f i , i = 1 , , . . . , n. ROBNER BASIS X β X β · · · X β n δ ◦ P ( D ) δ ◦ P ( D ) X β δ ◦ P ( D ) X β · · · δ ◦ P ( D ) X β n δ ◦ P ( D ) δ ◦ P ( D ) X β δ ◦ P ( D ) X β · · · δ ◦ P ( D ) X β n ... ... ... ... ... δ ◦ P n ( D ) δ ◦ P n ( D ) X β δ ◦ P n ( D ) X β · · · δ ◦ P n ( D ) X β n c c ... c n = f f ... f n . The coefficient matrix T ∆ is non-singular ⇔ The linear equations has a unique solu-tion.
Example
4. Given interpolation conditions ∆ = δ ◦ { , D y + D z , D x } and a setof monomials T = { , z, x } , then T ∆ = 1 z x ! δ ◦ { } δ ◦ { D y + D z } δ ◦ { D x } . It is obvious that T ∆ is non-singular, so T is an interpolation monomial basis for ∆. Lemma
Given interpolation conditions ∆ = δ { P ( D ) , P ( D ) , . . . , P n ( D ) } ,let T = { X β , X β , . . . , X β n } be an interpolation monomial basis for ∆ , then for each P i , ≤ i ≤ n , there exists X α i ∈ Λ { P i } satisfying X α i ∈ T .Proof. We will prove this by contradiction. Without loss of generality, we canassume that for every X α ∈ Λ { P } , X α T . It is observed that[ δ ◦ P ( D )] X β j = [ δ ◦ X ˆ P ( α ) D α ] X β j = X ˆ P ( α ) ( δ ◦ D α X β j ) | {z } = 0 , ≤ j ≤ n. So the matrix T ∆ has a zero row, and it is singular. It contradicts with Lemma 4.3. Remark δ ◦{ P ( D ) , P ( D ) , . . . , P n ( D ) } ,Lemma 4.4 shows that each P i , ≤ i ≤ n contains at least a monomial in the inter-polation monomial basis. Theorem
Given a monomial ordering ≺ and interpolation conditions ∆ = δ ◦ { P ( D ) , P ( D ) , . . . , P n ( D ) } , if { P , P , . . . , P n } is a “reverse” complete reducedbasis w.r.t. ≺ , then { lm( P ) , lm( P ) , . . . , lm( P n ) } is the ≺ -minimal monomial basisfor ∆ .Proof. According to Lemma 4.4, we must choose at least one monomial fromeach P i , ≤ i ≤ n to form the interpolation monomial basis. It is obvious that { lm( P ) , lm( P ) , . . . , lm( P n ) } we choose is of minimal degree w.r.t. ≺ . Thus weonly need to prove T = { lm( P ) , lm( P ) , . . . , lm( P n ) } do construct an interpolationmonomial basis for ∆, i.e., T ∆ is non-singular.Let P i = P ˆ P i ( α ) X α + ˆ P i ( β i ) X β i , lm( P i ) = X β i , ≤ i ≤ n . Since { P , P , . . . , P n } is a “reverse” complete reduced basis w.r.t. ≺ , it meanslm( P i ) Λ { P j } , i = j, ≤ i, j ≤ n. Y. H. GONG, X. JIANG
Hence we have( δ ◦ P j ( D ))(lm( P i )) = ( , i = j,β i ! ˆ P j ( β i ) = β i ! ˆ P i ( β i ) = 0 , i = j, ≤ i, j ≤ n. So T ∆ is a diagonal matrix with diagonal elements nonzero, i.e., it is non-singular.Since the ≺ -minimal monomial basis is equivalent [6] to the monomial basis ofquotient ring w.r.t. ≺ , we have the following theorem. Theorem
Given a monomial ordering ≺ and interpolation conditions ∆ = δ ◦ { P ( D ) , P ( D ) , . . . , P n ( D ) } , if { P , P , . . . , P n } is a “reverse” complete reducedbasis w.r.t. ≺ , then { lm( P ) , lm( P ) , . . . , lm( P n ) } is the monomial basis of quotientring F [ X ] /I (∆) w.r.t. ≺ . Example Given the monomial ordering grlex( z ≺ y ≺ x ) and interpolationconditions ∆ = δ ◦ { P ( D ) , P ( D ) , P ( D ) } = δ ◦ { , D y + D z , D x } . It is easy to see { P , P , P } = { , y + z, x } is a “reverse” complete reduced basisw.r.t. ≺ . Then by Theorem we know that { lm( P ) , lm( P ) , lm( P ) } = { , z, x } isthe monomial basis of quotient ring F [ X ] /I (∆) w.r.t. ≺ .
5. The algorithm to compute a Gr¨obner basis’s leading monomials.
Let ǫ = (1 , , . . . , , ǫ = (0 , , . . . , , . . . , ǫ d = (0 , , . . . , ∈ F d , for any β i ∈ F d , ≤ i ≤ n , we denote β i, ( j ) := β i + ǫ j , 1 ≤ i ≤ n, ≤ j ≤ d. Given a monomial ordering ≺ and interpolation conditions ∆ = δ ◦{ P ( D ) , P ( D ) , . . . , P n ( D ) } ,where { P , P , . . . , P n } is a “reverse” complete reduced basis w.r.t. ≺ , Algorithm 5.1yields leading monomials of the Gr¨obner basis for I (∆) w.r.t. ≺ . Algorithm 5.1
A Gr¨obner basis’s leading monomials Input:
A monomial ordering ≺ . The interpolation conditions ∆ = δ ◦ { P ( D ) , P ( D ) , . . . , P n ( D ) } , where { P , P , . . . , P n } is a “reverse” complete reduced basis w.r.t. ≺ , P i := P ˆ P i ( α ) X α + ˆ P i ( β i ) X β i , lm( P i ) = X β i , i = 1 , , . . . , n . Output: G , leading monomials of the Gr¨obner basis for I (∆). //Initialization QB := { X β , X β , . . . , X β n } , the monomial basis of quotient ring F [ X ] /I (∆). G := { X β , (1) , X β , (2) , . . . , X β , ( d ) , X β , (1) , X β , (2) , . . . , X β , ( d ) , . . . , X β n, (1) , X β n, (2) , . . . , X β n, ( d ) } . //Computing G := G − QB ; for i = 1 : n do for j = 1 : d do G := G − { multiples of X β i , (j) } ; G := G S { X β i, ( j ) } ; end for end for return G ;By theorem Theorem 4.7, it is easy to see Line 7 in Algorithm 5.1 is the monomialbasis of quotient ring F [ X ] /I (∆). Notice that the monomials in the quotient ring basisform a lower set [5], it is easy to check that Algorithm 5.1 is correct. ROBNER BASIS
6. The algorithm to compute a Gr¨obner basis.
Let X = ( x , x , . . . , x d ) and F [[ X ]] be the ring of formal power series. For θ = ( θ , θ , . . . , θ d ) ∈ F d , we denote by θ X := P di =1 θ i x i . From Taylor’s formula, wehave e θ X = ∞ X j =0 ( θ X ) j j ! . For any f ∈ F [[ X ]] we denote by λ n ( f ) the first finite terms of f with degrees ≤ n . Such ase θ X = 1 + ( θ X ) + ( θ X )
2! + · · · + ( θ X ) n n ! + ( θ X ) ( n +1) ( n + 1)! + · · · ∈ F [[ X ]] ,λ n (e θ X ) = 1 + ( θ X ) + ( θ X )
2! + · · · + ( θ X ) n n ! . Furthermore, by Taylor’s formula, we have(6.1) δ θ = δ ◦ e θ D . Since F [ X ] is isomorphic to F [[ X ]] [8], we get(6.2) δ θ ◦ { P ( D ) , P ( D ) , . . . , P n ( D ) } = δ ◦ { e θ D P ( D ) , e θ D P ( D ) , . . . , e θ D P n ( D ) } . It means that an interpolation problem at a nonzero point can be converted into oneat the zero point.
Algorithm 6.1
A Gr¨obner basis (Lagrange interpolation) Input:
A monomial ordering ≺ . The interpolation conditions ∆ = { δ θ , δ θ , . . . , δ θ n } , where distinct points θ i ∈ F d , i = 1 , , . . . , n . Output: { G , G , . . . , G m } , the reduced Gr¨obner basis for I (∆). //Initialization P := λ n (e θ X ) , P := λ n (e θ X ) , . . . , P n := λ n (e θ n X ). { P ( n )1 , P ( n )2 , . . . , P ( n ) n } , a “reverse” complete reduced basis w.r.t. ≺ , by Algo-rithm 3.1, P ( n ) j := P ˆ P ( n ) j ( α ) X α + ˆ P ( n ) j ( β j ) X β j , lm( P ( n ) j ) = X β j , j = 1 , , . . . , n . G = { X α , X α , . . . , X α m } , leading monomials of the Gr¨obner basis for I (∆), byAlgorithm 5.1 . G := X α , G := X α , . . . , G m := X α m . //Computing for i = 1 : m do G i := X α i − P nj =1 (cid:18) ( α i )! ˆ P ( n ) j ( α i )( β j )! ˆ P ( n ) j ( β j ) (cid:19) X β j ; end for return { G , G , . . . , G m } ;It is obvious that Algorithm 6.1 terminates. The following theorem shows itscorrectness. Y. H. GONG, X. JIANG
Theorem
The output { G , G , . . . , G m } in Algorithm is the reduced Gr¨obnerbasis for I (∆) w.r.t. ≺ .Proof. For an ideal interpolation problem with n interpolation conditions, sincethe monomials in the quotient ring basis form a lower set, and they appear in the leastmonomial of each polynomial in a “reverse” complete reduced basis (Theorem 4.7),it follows that we only need to compute the first finite terms of e θ X with degrees ≤ n in Line 6. By (6.1) and Theorem 4.7, it is easy to show that X α i ’s in Line13 are leading monomials of the reduced Gr¨obner basis and X β j ’s in Line 13 is thequotient ring basis. Now, the theorem will be proved if we can show that G i := X α i − P nj =1 (cid:18) ( α i )! ˆ P ( n ) j ( α i )( β j )! ˆ P ( n ) j ( β j ) (cid:19) X β j in Line 13 is in I (∆). Due to { P ( n )1 , P ( n )2 , . . . , P ( n ) n } in Line 7 is a “reverse” complete reduced basis, it follows that lm( P ( n ) j ) = X β j Λ { P ( n ) k } , j = k, ≤ j, k ≤ n . Hence, we have δ ◦ P ( n ) k ( D ) G i = δ ◦ P ( n ) k ( D )( X α i − n X j =1 ( α i )! ˆ P ( n ) j ( α i )( β j )! ˆ P ( n ) j ( β j ) ! X β j )= δ ◦ P ( n ) k ( D ) X α i − δ ◦ P ( n ) k ( D ) ( α i )! ˆ P ( n ) k ( α i )( β k )! ˆ P ( n ) k ( β k ) ! X β k = ( α i )! ˆ P ( n ) k ( α i ) − ( α i )! ˆ P ( n ) k ( α i )( β k )! ˆ P ( n ) k ( β k ) ! ( β k )! ˆ P ( n ) k ( β k )= 0 , ≤ k ≤ n, ≤ i ≤ m. Since { P k , ≤ k ≤ n } in Line 6 can be expressed linearly by { P ( n ) k , ≤ k ≤ n } in Line 7, it follows that δ ◦ P k ( D ) G i = 0 , ≤ k ≤ n, ≤ i ≤ m. i.e. δ ◦ λ n (e θ k D ) G i = 0 , ≤ k ≤ n, ≤ i ≤ m. Notice that G i ’s degree ≤ n , it is easy to see δ ◦ e θ k D G i = 0 , ≤ k ≤ n, ≤ i ≤ m. By (6.1), we have δ θ k G i = 0 , ≤ k ≤ n, ≤ i ≤ m. In other words, G i , ≤ i ≤ m vanish at θ k , ≤ k ≤ n . It follows that G i in Line 13is in I (∆), the proof is completed.Algorithm 6.1 shows that knowing the affine variety makes available informationconcerning the reduced Gr¨obner basis.We consider a bivariate example of ideal interpolation. Example
Given the monomial ordering grlex( y ≺ x ) and interpolation conditions ∆ = { δ (0 , , δ (1 , , δ (2 , } . We divide our computing of the reduced Gr¨obner basis for I (∆) in four steps. ROBNER BASIS (a) Interpolation points (b) Quotient ring basis Fig. 1 . Lagrange interpolation
First, we compute the polynomials in Line 6 of Algorithm , e (0 , X = 1 , e (1 , X = 1 + ( x + 2 y ) + 12! ( x + 2 y ) + 13! ( x + 2 y ) + · · · , e (2 , X = 1 + (2 x + y ) + 12! (2 x + y ) + 13! (2 x + y ) + · · · , we get { P , P , P } = { λ (1) , λ (e (1 , X ) , λ (e (2 , X ) } = { ,
13! ( x + 6 x y + 12 xy + 8 y ) + 12! ( x + 4 xy + 4 y ) + ( x + 2 y ) + 1 ,
13! (8 x + 12 x y + 6 xy + y ) + 12! (4 x + 4 xy + y ) + (2 x + y ) + 1 } . The next thing to do is computing a “reverse” complete reduced basis for { P , P , P } ,by Algorithm we get { P (3)1 , P (3)2 , P (3)3 } = { , ( − x + 2 xy + 53 y ) + ( − x + 43 xy + 73 y ) + (2 y ) , ( 56 x + x y − y ) + ( 76 x + 23 xy − y ) + ( x ) } . Another step is computing the reduced Gr¨obner basis’s leading monomials, byAlgorithm we get QB = { lm( P (3)1 ) , lm( P (3)2 ) , lm( P (3)3 ) } = { , y, x } is the monomial basis of quotient ring F [ X ] /I (∆) , and G = { y , xy, x } are leading monomials of the reduced Gr¨obner basis for I (∆) .Finally, we compute the reduced Gr¨obner basis for I (∆) , by Line 13 in Algo-rithm we get Y. H. GONG, X. JIANG G = y − y + 23 x,G = xy − y − x,G = x + 23 y − x, { G , G , G } = { y + x − y, xy − x − y, x − x + y } is the reduced Gr¨obnerbasis for I (∆) w.r.t. ≺ . By (6.2), proceeding as in the proof of Theorem 6.1, we have
Algorithm 6.2
A Gr¨obner basis (Hermite interpolation) Input:
A monomial ordering ≺ . The interpolation conditions∆ = δ θ ◦ { P ( D ) , P ( D ) , . . . , P s ( D ) } δ θ ◦ { P ( D ) , P ( D ) , . . . , P s ( D ) } ... δ θ k ◦ { P k ( D ) , P k ( D ) , . . . , P ks k ( D ) } , where distinct points θ i ∈ F d , i = 1 , , . . . , k and s + s + · · · + s k = n. Output: { G , G , . . . , G m } , the reduced Gr¨obner basis for I (∆). //Initialization P := λ n (e θ X P ) , P := λ n (e θ X P ) , . . . , P s := λ n (e θ X P s ) , P s +1 := λ n (e θ X P ) , . . . , P n := λ n (e θ k X P ks k ). { P ( n )1 , P ( n )2 , . . . , P ( n ) n } , a “reverse” complete reduced basis w.r.t. ≺ , by Algo-rithm 3.1, P ( n ) j := P ˆ P ( n ) j ( α ) X α + ˆ P ( n ) j ( β j ) X β j , lm( P ( n ) j ) = X β j , j = 1 , , . . . , n . G = { X α , X α , . . . , X α m } , leading monomials of the Gr¨obner basis for I (∆), byAlgorithm 5.1 . G := X α , G := X α , . . . , G m := X α m . //Computing for i = 1 : m do G i := X α i − P nj =1 (cid:18) ( α i )! ˆ P ( n ) j ( α i )( β j )! ˆ P ( n ) j ( β j ) (cid:19) X β j ; end for return { G , G , . . . , G m } ; Example
Given the monomial ordering grlex( y ≺ x ) and interpolation conditions ∆ = (cid:26) δ (0 , ◦ { , D x , D x + D y } δ (1 , ◦ { , D x } , We divide our computing of the reduced Gr¨obner basis for I (∆) in four steps.First, we compute the polynomials in Line 6 of Algorithm , ROBNER BASIS (a) Interpolation points (b) Quotient ring basis Fig. 2 . Hermite interpolation e (0 , X = 1 , e (1 , X = 1 + ( x + 2 y ) + 12! ( x + 2 y ) + 13! ( x + 2 y ) + · · · , we get { P , P , P , P , P } = { λ (1) , λ ( x ) , λ ( 12 x + y ) , λ (e (1 , X ) , λ (e (1 , X x ) } = { , x, x + y,
15! ( x + 2 y ) + 14! ( x + 2 y ) + 13! ( x + 2 y ) + 12! ( x + 2 y ) + ( x + 2 y ) + 1 ,
14! ( x + 2 y ) x + 13! ( x + 2 y ) x + 12! ( x + 2 y ) x + ( x + 2 y ) x + x } . The next thing to do is computing a “reverse” complete reduced basis for { P , P , P , P , P } ,by Algorithm we get { P (5)1 , P (5)2 , P (5)3 , P (5)4 , P (5)5 } = { , x, x + y, ( 15! ( x + 2 y ) −
14! ( x + 2 y ) x ) + ( 14! ( x + 2 y ) −
13! ( x + 2 y ) x ) + ( − x − x y + 43 y ) + ( − x + 2 y ) ,
14! ( x + 2 y ) x + 13! ( x + 2 y ) x + ( 12 x + 2 x y + 2 xy ) + ( x + 2 xy ) } . Another step is computing the reduced Gr¨obner basis’s leading monomials, byAlgorithm we get QB = { lm( P (5)1 ) , lm( P (5)2 ) , lm( P (5)3 ) , lm ( P (5)4 ) , lm ( P (5)5 ) } = { , x, y, y , xy } is the monomial basis of quotient ring F [ X ] /I (∆) , and G = { y , xy , x } are leading monomials of the reduced Gr¨obner basis for I (∆) .Finally, we compute the reduced Gr¨obner basis for I (∆) , by Line 13 in Algo-rithm we get Y. H. GONG, X. JIANG G = y − y ,G = xy − xy,G = x − y + 34 y − xy, { G , G , G } = { y − y , xy − xy, x − xy + y − y } is the reduced Gr¨obner basisfor I (∆) w.r.t. ≺ .
7. Conclusions.
Given a monomial ordering ≺ and ideal interpolation conditionfunctionals ∆, by the concept of “reverse” complete reduced basis, we give a methodto find the reduced Gr¨obner basis for I (∆) w.r.t. ≺ . Our algorithm for “reverse”complete reduced basis only uses linear eliminating, so it has a good performance. REFERENCES[1] G. Birkhoff, The algebra of multivariate interpolation. In: Coffman, C.V., Fix, G.J. (eds.)Constructive Approaches to Mathematical Models, pp. 345-363. Academic Press, New York(1979).[2] B. Buchberger, Ein algorithmus zum auffinden der basiselemente des restklassenrings nach einemnulldimensionalen polynomideal, Ph.D. thesis, Innsbruck, 1965.[3] H. M¨oller, B. Buchberger, The construction of multivariate polynomials with preassigned zeros,in: J. Calmet(Ed.), Computer Algebra: EUROCAM ’82. in: Lecture Notes in ComputerScience, vol. 144, Springer, Berlin, 1982. pp. 24-31.[4] M. G. Marinari, H. M. M¨oller, and T. Mora, Gr¨obner bases of ideals defined by functionals with anapplication to ideals of projective points, Applicable Algebra in Engineering Communicationand Computing. 4(1993), pp. 103–145.[5] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms- An Introduction to Compu-tational Algebraic Geometry and Commutative Algebra, Springer, New York, 3rd ed., 2007.[6] T. Sauer, Polynomial interpolation of minimal degree and Gr¨obner bases, in Gr¨obner Bases andApplications (Proc. of the Conf.33 Years of Gr¨obner Bases), Cambridge University Press,pp. 483–494.[7] C. de Boor, Ideal interpolation. In: Chui, C.K., Neamtu, M., Schumaker, L. (eds.) ApproximationTheory XI: Gatlinburg 2004, pp. 59-91. Nashboro Press, Brentwood (2005).[8] C. de Boor, A. Ron, On multivariate polynomial interpolation. Constr. Approx.6