aa r X i v : . [ m a t h . A C ] J a n A Two-Dimensional Improvement for Farr-GaoAlgorithm ∗ Tian Dong † School of Mathematics, Key Lab. of Symbolic Computationand Knowledge Engineering (Ministry of Education),Jilin University, Changchun 130012, China
Abstract
Farr-Gao algorithm is a state-of-the-art algorithm for reduced Gr¨obnerbases of vanishing ideals of finite points, which has been implementedin
Maple r as a build-in command. In this paper, we present a two-dimensional improvement for it that employs a preprocessing strategy forcomputing reduced Gr¨obner bases associated with tower subsets of givenpoint sets. Experimental results show that the preprocessed Farr-Gaoalgorithm is more efficient than the classical one. Keywords : Gr¨obner basis; Vanishing ideal; Tower set; Gr¨obner ´escalier;Newton interpolation basis : 13P10
Let F be a field, and let Π d := F [ x , x , . . . , x d ] denote the d -variate polynomialring over F . It is well known that the set of polynomials in Π d that vanish at afinite nonempty set Ξ ⊂ F d forms an ideal in Π d which is called the vanishingideal of Ξ, denoted by I (Ξ). In view of the applications of vanishing ideals inthe fields of mathematics and other sciences in recent years[6, 8], there has beenincreasing interest in their Gr¨obner bases[1] and Gr¨obner ´escaliers (aka standardset, standard monomials etc.)[12].The most significant milestone of computing vanishing ideals is the Buch-berger -M¨oller algorithm [11] that yields, for fixed Ξ and monomial order ≺ onΠ d , the reduced Gr¨obner basis G and the Gr¨obner ´escalier M of I (Ξ) w.r.t. ≺ .It also produces a Newton interpolation basis N for the F -linear space spannedby M . One decade later, MMM algorithm in [10] extended Buchberger-M¨olleralgorithm to solve general zero-dimensional ideals. And then, [7] introduced a ∗ Supported by National Natural Science Foundation of China (No. 11101185). † email: [email protected] Q d .All these three algorithms apply Gauss elimination on generalized Vandermondematrices regardless of the order of the points in Ξ.As is well known, G depends not only on F d (see [14]).However, the algorithms mentioned above do not take this into account. In 2006,Farr and Gao [5] presented a more effcient algorithm for G (called Farr-Gao al-gorithm hereafter) that has been implemented in Maple r as build-in command VanishingIdeal . Arguably, the most distinguishing feature of Farr-Gao algo-rithm is its point-sorting strategy that provides the possibility to borrow theidea of univariate Newton interpolation. Once the points are sorted, the compu-tation will be performed along parallel lines, one after another. On each line, weare essentially solving univariate Newton interpolation and hence the amount ofreduction is decreased. The process of Farr-Gao algorithm implies that multi-variate Newton interpolation would be helpful for the computation of vanishingideals. Concretely, if we could theoretically obtain a Newton interpolation basisof some subset of Ξ, then the amount of reduction of Farr-Gao algorithm wouldbe decreased further.Let Ξ be a Cartesian set in F d . [13] gave the unique Gr¨obner ´escalier M of I (Ξ) in theory which implies that I (Ξ) has a unique reduced Gr¨obner basesw.r.t. any monomial order (see [9]). Moreover, we can also construct Newtoninterpolation bases for Ξ theoretically. Based on this, [15] proposed a bivariatepreprocessing paradigm for Buchberger-M¨oller algorithm that inputs the mono-mial (Gr¨obner ´escalier) and Newton interpolation basis for a maximal Cartesiansubset of Ξ into Buchberger-M¨oller algorithm as initial values. However, sincethe distribution of a Cartesian set is fairly restricted, in many cases the maximalCartesian subsets are not large enough and therefore the improvement is minor.In the following, we first introduce a new type of finite nonempty sets, towersets, in F that have “freer” distributions than bivariate Cartesian sets whoseformal definition is provided in Section 2 where we also establish a new crite-rion for bivariate Cartesian sets for the purpose of investigating the relationbetween tower sets and Cartesian sets. Next, in Section 3, we theoretically offerthe Gr¨obner ´escaliers of vanishing ideals of tower sets w.r.t. commonly usedmonomial orders as well as the Newton interpolation bases spanned by them.And, finally, these results lead to our main algorithm and the timings of someexperiments are given. Let N stand for the monoid of nonnegative integers. A polynomial f ∈ Π = F [ x, y ] is of the form f = X ( i,j ) ∈ N c ij x i y j , { ( i, j ) ∈ N : 0 = c ij ∈ F } < ∞ , monomial x i y j is a product for vector ( i, j ). The set of all monomials inΠ is denoted by T .Fix a monomial order ≺ on Π that could be lexicographical order ≺ lex ( plex(x, y) in Maple r ), inverse lexicographical order ≺ inlex ( plex(y, x) in Maple r ), graded lexicographical order ≺ grlex ( grlex(x, y) in Maple r ), orgraded reverse lexicographical order ≺ grevlex ( tdeg(x, y) in Maple r ) etc, cf.[1]. For all nonzero f ∈ Π , we let LT ≺ ( f ) signify the leading term , LM ≺ ( f )the leading monomial , and LC ≺ ( f ) the leading coefficient of f . Furthermore,for a nonempty set F ⊂ Π , setLM ≺ ( F ) := { LM ≺ ( f ) : f ∈ F } . Let G be the reduced Gr¨obner basis for a zero-dimensional ideal I ⊂ Π w.r.t. ≺ . According to [12], the monomial set N ≺ ( I ) := { t ∈ T : LM ≺ ( g ) ∤ t, ∀ g ∈ G } (1)forms the Gr¨obner ´escalier of I w.r.t. ≺ , and its corner C [ N ≺ ( I )] := (cid:8) t ∈ T : x | t ⇒ t/x ∈ N ≺ ( I ) , y | t ⇒ t/y ∈ N ≺ ( I ) (cid:9) \N ≺ ( I ) (2)is equal to LM ≺ ( G ).Let A be a finite nonempty set in N . It is called lower if for any ( i, j ) ∈ A we always have { ( i ′ , j ′ ) ∈ N : 0 ≤ i ′ ≤ i, ≤ j ′ ≤ j } ⊆ A . (3)Set b x ( A ) := max ( i, ∈A i, b y ( A ) := max (0 ,j ) ∈A j. (4)Subfigure (b) of Fig. 1 illustrates a lower set with (b x , b y ) = (4 ,
7) labeled. Ob-viously, b x ( A ) and b y ( A ) alone are not enough to determine A . Hence, we in-troduce sequences m , m , . . . , m b y ( A ) and n , n , . . . , n b x ( A ) that can uniquelydetermine A respectively, where m j = max ( i,j ) ∈A i, ≤ j ≤ b y ( A ) ,n i = max ( i,j ) ∈A j, ≤ i ≤ b x ( A ) . It is easy to see that (4) implies m = b x ( A ) and n = b y ( A ). Thus, it makessense to write A = L x ( m , m , . . . , m b y ( A ) ) = L y ( n , n , . . . , n b x ( A ) ) . (5)A simple observation shows that the lower set in Subfigure (b) of Fig. 1 isL x (4 , , , , , , ,
0) = L y (7 , , , , . m , . . . , m b y ( A ) and n , . . . , n b x ( A ) are monotonically decreasing sequences. Furthermore, if they are strictly mono-tonically decreasing, then we say that A is x - strict (resp. y - strict ) lower.As index sets, the lower sets in N are used to label Cartesian sets in F asfollows. Definition 2.1 [9]
A finite nonempty set Ξ ⊂ F of distinct points is Cartesian if and only if there exists a lower set
A ⊂ N and two injective functions x , y : N → F such that Ξ can be written as Ξ = (cid:8) (x( i ) , y( j )) ∈ F : ( i, j ) ∈ A (cid:9) . (6)Ξ is also called A - Cartesian . Subfigure (a) of Fig. 1 illustrates a Cartesian set that is labelled by the lowerset mentioned above.Given a finite nonempty set Ξ ⊂ F of distinct points. Set π x (Ξ) := { π x ( ξ ) : ξ ∈ Ξ } and π y (Ξ) := { π y ( ξ ) : ξ ∈ Ξ } as the first and the second projection maps on F respectively, namely π x : F → F : ( x, y ) x and π y : F → F : ( x, y ) y. Recall the point-sorting strategy of Farr-Gao algorithm, Ξ can be decom-posed vertically and horizontally asΞ = [ ¯ x ∈ π x (Ξ) Ξ ∩ { x = ¯ x } =: π x (Ξ) − [ i =0 Ξ yi = [ ¯ y ∈ π y (Ξ) Ξ ∩ { y = ¯ y } =: π y (Ξ) − [ j =0 Ξ xj , (7)where y ≥ · · · ≥ y π x (Ξ) − and x ≥ · · · ≥ x π y (Ξ) − . Subfigure (a)of Fig.1 displays the decompositions of a Cartesian set.In [2], two particular lower sets in N S x (Ξ) :=L x ( x − , . . . , x π y (Ξ) − − ,S y (Ξ) :=L y ( y − , . . . , y π x (Ξ) − −
1) (8)are constructed from Ξ (see (b) of Fig.1 for example), which reflect the distri-bution of Ξ in certain sense, and the following criterion for Cartesian sets in F is offered as well. Theorem 2.1 [2]
A finite nonempty set Ξ ⊂ F is Cartesian if and only if S x (Ξ) = S y (Ξ) . y y Ξ y Ξ y Ξ y Ξ y Ξ x Ξ x Ξ x Ξ x Ξ x Ξ x Ξ x Ξ x (a) Ξ = S i =0 Ξ yi = S j =0 Ξ xj mn , (b) S y (Ξ) Figure 1: Cartesian set Ξ, its decompositions, and S y (Ξ)Theorem 2.1, Definition 2.1, and (8) yield the following corollary immedi-ately. Corollary 2.1
If a finite nonempty set Ξ ⊂ F is A -Cartesian, then A = S x (Ξ) = S y (Ξ) . Consequently, (6) can be rewritten as Ξ = (cid:8) (x( i ) , y( j )) ∈ F : ( i, j ) ∈ S x (Ξ) (cid:9) (9)= (cid:8) (x( i ) , y( j )) ∈ F : ( i, j ) ∈ S y (Ξ) (cid:9) . (10)Unfortunately, it is difficult to extend Theorem 2.1 to three and higher di-mensions. Therefore, we give the following criterion that extends to higherdimensions naturally. Theorem 2.2
A finite nonempty set Ξ ⊂ F with decompositions (7) is Carte-sian if and only if π x (Ξ x ) ⊇ π x (Ξ x ) ⊇ · · · ⊇ π x (Ξ x π y (Ξ) − ) (11) or π y (Ξ y ) ⊇ π y (Ξ y ) ⊇ · · · ⊇ π y (Ξ y π x (Ξ) − ) . (12) Proof:
Assume that Ξ is A -Cartesian satisfying (6) where A can be repre-sented as (5) that together with (6) implies b y ( A ) = π y (Ξ) − π x (Ξ xj ) = { x(0) , x(1) , . . . , x( m j ) } , j = 0 , . . . , b y ( A ) . Since m ≥ m ≥ · · · ≥ m b y ( A ) ≥
0, (11) follows. (12) can be proved in likemanner. 5onversely, we suppose that (11) holds and that lower sets S x (Ξ) and S y (Ξ)have the expression (8). Then there exists a unique sequence ( n , n , . . . , n x − ) ∈ N x such that S x (Ξ) = L y ( n , n , . . . , n x − ) (13)where n = π y (Ξ) − π x (Ξ) = x , namely we can cover Ξ by exactly x vertical lines. Prove this by contradiction. It is evident from π x (Ξ) ≥ x that the equality fails only when π x (Ξ) > x , i.e., there exists at leastone point (¯ x, ¯ y ) ∈ Ξ such that ¯ x / ∈ π x (Ξ x ). However, (¯ x, ¯ y ) ∈ Ξ simply meansthat there exists some 0 < j ≤ π y (Ξ) − x, ¯ y ) ∈ Ξ xj . Thus (11)implies that ¯ x ∈ π x (Ξ xj ) ⊆ π x (Ξ x ) which contradicts ¯ x / ∈ π x (Ξ x ).In the rest of the proof, we will use induction on h to show that n h = yh − , h = 0 , . . . , π x (Ξ) − , (14)which leads to S x (Ξ) = S y (Ξ) immediately. When h = 0, for every (¯ x, ¯ y ) ∈ Ξ x π y (Ξ) − , it follows from (11) that ¯ x ∈ π x (Ξ x π y (Ξ) − ) ⊂ π x (Ξ xj ) , ≤ j ≤ π y (Ξ) −
2, which means that { Ξ ∩ { x = ¯ x }} ≥ π y (Ξ). But { Ξ ∩ { x =¯ x }} ≤ π y (Ξ) is trivial, we have y = { Ξ ∩ { x = ¯ x }} = π y (Ξ) = n + 1,namely (14) is true for h = 0.Now assume (14) for 0 ≤ h ≤ k < π x (Ξ) −
1, i.e., n h = yh − , h =0 , . . . , k . It turns out that there exist distinct ¯ x , ¯ x , . . . , ¯ x k ∈ π x (Ξ) such that { Ξ ∩ { x = ¯ x h }} = n h + 1 , ≤ h ≤ k . Thus, for every 0 ≤ h ≤ k , n h ≥ n k +1 implies ¯ x h ∈ π x (Ξ xn k +1 ). Since ( k + 1 , n k +1 ) ∈ S x (Ξ), there exists atleast one point (¯ x k +1 , ¯ y k +1 ) ∈ Ξ xn k +1 that is not in Ξ yj , j = 0 , . . . , k . By (11),a similar argument leads to { Ξ ∩ { x = ¯ x k +1 }} ≥ n k +1 + 1 which implies yk +1 ≥ n k +1 + 1. On the other side, it follows from induction hypothesisthat every point in Ξ yk +1 belongs to some Ξ xj , ≤ j ≤ n k +1 , which implies that yk +1 ≤ n k +1 + 1, therefore we have yk +1 = n k +1 + 1, namely (14) holds for h = k + 1. Theorem 2.1 immediately implies that Ξ is Cartesian.Swapping the roles of x and y , the other statement can be proved similarly. (cid:3) As mentioned in Section 1, [13] provides the Gr¨obner ´escalier of the vanishingideal of a Cartesian set in theory. In view of a later application, we restate theresult only in case d = 2. Theorem 2.3 [13]
Let Ξ ⊂ F be an A -Cartesian set. Then Gr¨obner ´escalier N ≺ ( I (Ξ)) w.r.t. any monomial order ≺ is identical to M A := { x i y j : ( i, j ) ∈ A} . (15)Theorem 2.3 indicates that an A -Cartesian set in F has the advantagethat the Gr¨obner ´escalier of its vanishing ideal can be easily obtained from thestructure of A . Nevertheless, Theorem 2.2 illustrates that the distribution ofa Cartesian set in F is highly restricted. Naturally, we wonder if there existsanother type of finite nonempty sets with “freer” distribution and (15)-likeproperty. 6 efinition 2.2 Keep the notation above. A finite nonempty set Ξ in F istermed x - tower (resp. y - tower ) if S x (Ξ) is x -strict ( resp. S y (Ξ) is y -strict ) andthere exist two injective functions x , y : N → F as well as b y ( S x (Ξ)) + 1 ( resp. b x ( S y (Ξ)) + 1) permutations σ x , . . . , σ x b y ( S x (Ξ)) ( resp. σ y , . . . , σ y b x ( S y (Ξ)) ) of set { , , . . . , b x ( S x (Ξ)) } ( resp. { , , . . . , b y ( S y (Ξ)) } ) such that Ξ can be written as Ξ := { (x( σ xj ( i )) , y( j )) : ( i, j ) ∈ S x (Ξ) } (16)( resp. Ξ := { (x( i ) , y( σ yj ( j ))) : ( i, j ) ∈ S y (Ξ) } ) . (17)Fix the injective functions x and y. Comparing (16) with (9), we find thatif the permutations in (16) are all identical, then (16) is same as (9) in form.Assume that ( i, j ) ∈ S x (Ξ). If Ξ ⊂ F is Cartesian, by (9), the correspondingpoint of ( i, j ) in Ξ must be (x( i ) , y( j )). But when Ξ is x -tower, since σ xj isarbitrary, the corresponding point of ( i, j ) could be any one of (x( h ) , y( j )) , h =0 , . . . , b x ( S x (Ξ)). Symmetrically, a y -tower set has the same behavior in verticaldirection. Then b x ( S x (Ξ)) = x − y ( S y (Ξ)) = y − Theorem 2.4
A finite nonempty set Ξ ⊂ F with decompositions (7) is x -tower ( resp. y -tower ) if and only if S x (Ξ) is x -strict ( resp. S y (Ξ) is y -strict ) and π x (Ξ x ) ) π x (Ξ xj ) , j = 1 , , . . . , π y (Ξ) − resp. π y (Ξ y ) ) π y (Ξ yi ) , i = 1 , , . . . , π x (Ξ) − . Subfigure (a) of Fig. 2 illustrates an x -tower set Ξ with lower set S x (Ξ) =(11 , , , , , x -tower (resp. y -tower) set is “freer”than a Cartesian set. Nonetheless, when it comes to the number of the pointson each line, Cartesian sets are winners this time, because their S x (Ξ)(= S y (Ξ))are not restricted to be x -strict or y -strict.By Theorem 2.2, a tower set Ξ ⊂ F becomes a Cartesian set if and onlyif (11) or (12) is satisfied. Conversely, it follows from Theorem 2.4 that an A -Cartesian set Ξ ⊂ F is x -tower(resp. y -tower) if and only if A is x -strict (resp. y -strict). Consequently, it turns out that the notions of Cartesian set and towerset in F are not mutually exclusive. Nevertheless, Theorem 2.2 and 2.4 alsoimplies that most tower sets are not Cartesian and vice versa. For example, setΞ in (a) of Fig. 2 is x -tower but not Cartesian while set Ξ ′ in (b) of Fig. 2 isCartesian but not x -tower or y -tower. We need the following lemma and definition before we give Theorem 3.1 that the-oretically provides the Gr¨obner ´escalier of the vanishing ideal of an x -tower(resp.7 y x Ξ x Ξ x Ξ x Ξ x Ξ x (a) Ξ xy x Ξ x Ξ x Ξ x Ξ x Ξ x (b) Ξ ′ Figure 2: A non-Cartesian tower set and a non-tower Cartesian set y -tower) set in F w.r.t. ≺ grlex (resp. ≺ grevlex ). Lemma 3.1 [4]
Let
Ξ = { ( x , y ) , ( x , y ) , . . . , ( x m , y ) } ⊂ F be a set of m + 1 distinct points on line y = y . Then I (Ξ) = h ( x − x )( x − x ) · · · ( x − x m ) , y − y i . Definition 3.1 [1]
Fix a monomial order ≺ and let F = { f , . . . , f s } ⊂ Π d with f i = 0 . Given f, f ′ ∈ Π d , we say that f reduces to f ′ modulo f in one step,written f f −→ f ′ if and only if LT ≺ ( f ) divides a nonzero term t that appears in f and f ′ = f − tLT ≺ ( f ) f . Moreover, we say that f reduces to f ′ modulo F , denoted f F −→ + f ′ , if and only if there exist a sequence of indices i , . . . , i t ∈ { , . . . , s } and asequence of polynomials h , . . . , h t − ∈ Π d such that f f i −−→ h f i −−→ h f i −−→ · · · f it − −−−→ h t − f it −−→ f ′ . Theorem 3.1
Given an x -tower ( resp. y -tower ) set Ξ ⊂ F . The Gr¨obner´escalier of vanishing ideal I (Ξ) w.r.t. ≺ grlex ( resp. ≺ grevlex ) is M S x (Ξ) ( resp. M S y (Ξ) ) . roof: We only offer the proof of the first statement. The second one can beverified in very like fashion.Retain all the notation established previously. Fix monomial order as ≺ grlex ,and for simplicity the symbol will be omitted in the rest of the proof where noconfusion arises. Suppose that x -tower set Ξ has the decompositions (7). Forconvenience, we set m j := xj − , j = 0 , . . . , ν , where ν = π y (Ξ) − y ( S x (Ξ)). Thus, (8) implies S x (Ξ) = L x ( m , m , . . . , m ν ). Fix 0 ≤ j ≤ ν . ByLemma 3.1, the ideal I j := I (Ξ xj ) = I ( { (x( σ xj ( i )) , y( j )) : ( i, j ) ∈ S x (Ξ) } )= * Y ( i,j ) ∈ S x (Ξ) ( x − x( σ xj ( i ))) , y − y( j ) + . Obviously, I j ’s are pairwise comaximal. Hence I (Ξ) = ν \ j =0 I j = ν Y j =0 I j = ν Y j =0 * Y ( i,j ) ∈ S x (Ξ) ( x − x( σ xj ( i ))) , y − y( j ) + . Let G N be the reduced Gr¨obner basis for ideal Q Nj =0 I j w.r.t. ≺ grlex , 0 ≤ N ≤ ν .We will use induction on N to proveLM( G N ) = { x m +1 , x m +1 y, . . . , x m N +1 y N , y N +1 } , N = 0 , , . . . , ν. (18)First of all, LM Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) = x x = x m +1 leads to (18) immediately for N = 0.It follows from Theorem 2.4 that m > m and π x (Ξ x ) ( π x (Ξ x ). Therefore,we have I · I = * Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) , y − y(0) + · * Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) , y − y(1) + = * Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) , Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! ( y − y(1)) , Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! ( y − y(0)) , Y j =0 ( y − y( j )) + = * Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) , Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! ( y − y(0)) , Y j =0 ( y − y( j )) + Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! ( y − y(1)) . Then LM( G ) = { x m +1 , x m +1 y, y } follows which means that (18) holds for N = 1.Similarly, by m > m and π x (Ξ x ) ( π x (Ξ x ), we obtain, after some easycomputations, I · I · I = * Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) , Y j =0 ( y − y( j )) , Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! ( y − y(0)) , Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! ( y − y(0))( y − y(2)) , Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! Y j =0 ( y − y( j )) + =: D g (2)0 , g (2)1 , g (2)2 , g (2)3 , g (2)4 E . We let ˆ q, ˆ r ∈ Π be the quotient and the remainder of the division of Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) by Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) respectively, namely Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) = ˆ q Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! + ˆ r. Denote the remainder of g (2)3 w.r.t. g (2)4 by ¯ g (2)3 . One can check readily that¯ g (2)3 = g (2)3 − ˆ qg (2)4 . On the other hand, g (2)2 = 1y(1) − y(2) " Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! g (2)3 − Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! g (2)4 g (2)2 { g (2)3 ,g (2)4 } −−−−−−−→ +
0. Consequently, I · I · I = D g (2)0 , g (2)1 , ¯ g (2)3 , g (2)4 E . We claim that G = G ′ := n g (2)0 , g (2)1 , (y(1) − y(2)) − ¯ g (2)3 , g (2)4 o , where y(1) − y(2) = LC (cid:16) ¯ g (2)3 (cid:17) , i.e., (y(1) − y(2)) − ¯ g (2)3 is monic.In fact, if S ( f, g ) stands for the S-polynomial of polynomials f, g ∈ Π , then S (cid:16) g (2)0 , g (2)1 (cid:17) G ′ −−→ + (cid:16) g (2)0 (cid:17) and LM (cid:16) g (2)1 (cid:17) are relatively prime. Observing that Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) is a factor of Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))), we get S (cid:16) g (2)0 , g (2)4 (cid:17) = y g (2)0 − x m − m g (2)4 = y g (2)0 − x m − m g (2)4 − Y j =0 ( y − y( j )) ! g (2)0 + Y j =0 ( y − y( j )) ! g (2)0 = y g (2)0 − x m − m g (2)4 − Y j =0 ( y − y( j )) ! g (2)0 + Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) g (2)4 =((y(0) + y(1)) y − y(0)y(1)) g (2)0 + Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) − x m − m ! g (2)4 . It is easy to see that LM (cid:16) S (cid:16) g (2)0 , g (2)4 (cid:17)(cid:17) = x m +1 y . Moreover, a simple com-putation leads to: LM (cid:16) ((y(0) + y(1)) y − y(0)y(1)) g (2)0 (cid:17) = x m +1 y, LM Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) − x m − m ! g (2)4 ! = x m y , therefore max ≺ grlex LM (cid:16) ((y(0) + y(1)) y − y(0)y(1)) g (2)0 (cid:17) , LM Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) Q ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) − x m − m ! g (2)4 ! ! = x m +1 y = LM (cid:16) S (cid:16) g (2)0 , g (2)4 (cid:17)(cid:17) S (cid:16) g (2)0 , g (2)4 (cid:17) G ′ −−→ +
0. Similarly, S (cid:16) g (2)1 , g (2)4 (cid:17) = x m +1 − Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! g (2)1 − y(2) g (2)4 and LM (cid:16) S (cid:16) g (2)1 , g (2)4 (cid:17)(cid:17) = x m +1 y implymax ≺ grlex LM x m +1 − Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ! g (2)1 ! , LM (cid:16) − y(2) g (2)4 (cid:17) ! = max ≺ grlex (cid:0) x m y , x m +1 y (cid:1) = x m +1 y = LM (cid:16) S (cid:16) g (2)1 , g (2)4 (cid:17)(cid:17) , which means that S (cid:16) g (2)1 , g (2)4 (cid:17) G ′ −−→ + S (cid:16) g (2)0 , (y(1) − y(2)) − ¯ g (2)3 (cid:17) G ′ −−→ + , S (cid:16) g (2)1 , (y(1) − y(2)) − ¯ g (2)3 (cid:17) G ′ −−→ + . Hence, there only remains S (cid:16) (y(1) − y(2)) − ¯ g (2)3 , g (2)4 (cid:17) G ′ −−→ + S (cid:16) (y(1) − y(2)) − ¯ g (2)3 , g (2)4 (cid:17) = ˆ r y(1) − y(2) g (2)1 + y(1) · (y(1) − y(2)) − ¯ g (2)3 + (cid:0) ˆ q − x m − m (cid:1) g (2)4 whose leading monomial is x m +1 y . ThusLM (cid:18) ˆ r y(1) − y(2) g (2)1 (cid:19) = x m y , LM (cid:16) y(1) · (y(1) − y(2)) − ¯ g (2)3 (cid:17) = x m +1 y, and LM (cid:16)(cid:0) ˆ q − x m − m (cid:1) g (2)4 (cid:17) = x m y imply that S (cid:16) (y(1) − y(2)) − ¯ g (2)3 , g (2)4 (cid:17) G ′ −−→ + S ( g, g ′ ) G ′ −−→ + g, g ′ ∈ G ′ . By Buchberger’s S-pair criterion, G ′ is aGr¨obner basis for I · I · I w.r.t. ≺ grlex . Moreover, for every g ∈ G ′ , it isevident that1. LC( g ) = 1,2. No monomial of g lies in h LT( G ′ − { g } ) i ,12hich means that G ′ is reduced, namely (18) holds for N = 2.Now, assume (18) for N = k, ≤ k < ν . Without loss of generality, wesuppose that G k = n g ( k )0 , . . . , g ( k ) k +1 o withLM (cid:16) g ( k ) i (cid:17) = x m i +1 y i , i = 0 , . . . , k, LM (cid:16) g ( k ) k +1 (cid:17) = y k +1 , which imply that g ( k )0 = Y ( i, ∈ S x (Ξ) ( x − x( σ x ( i ))) ,g ( k ) k +1 = k Y j =0 ( y − y( j )) . When N = k + 1, since Ξ is x -tower, we obtain k +1 Y j =0 I j =( k Y j =0 I j ) I k +1 = D g ( k )0 , . . . , g ( k ) k +1 E · * Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) , y − y( k + 1) + = * g ( k +1)0 , g ( k )0 ( y − y( k + 1)) ,g ( k )1 Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! , g ( k )1 ( y − y( k + 1)) , · · · g ( k ) k − Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! , g ( k ) k − ( y − y( k + 1)) ,g ( k ) k Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! , g ( k ) k ( y − y( k + 1)) ,g ( k ) k +1 Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! , g ( k ) k +1 ( y − y( k + 1)) + , where g ( k +1)0 = g ( k )0 is obvious hence g ( k )0 ( y − y( k + 1)) can be removed. By theinduction hypothesis, we have g ( k +1) k +2 := g ( k ) k +1 ( y − y( k + 1)) = k +1 Y j =0 ( y − y( j )) .
13e denote polynomial g ( k ) k +1 ( Q ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i )))) by g ( k +1) k +1 . It followsfrom the induction hypothesis that LM (cid:16) g ( k +1) k +1 (cid:17) = x m k +1 +1 y k +1 . Set E := n g ( k +1) k +1 o . Suppose that g ( k ) k ( y − y( k + 1)) E −−→ + g ( k +1) k . Since m k > m k +1 , wehave LM (cid:16) g ( k +1) k (cid:17) = x m k +1 y k . Recall case N = 2. It is easy to see that g ( k ) k Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! F −→ + F := n g ( k +1) k +1 , g ( k +1) k o , which means that g ( k ) k ( Q ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))))can be removed from the original ideal basis.Set E := F , and suppose that g ( k ) k − ( y − y( k + 1)) E −−→ + g ( k +1) k − . We similarlydeduce that LM (cid:16) g ( k +1) k − (cid:17) = x m k − +1 y k − and g ( k ) k − Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! F −→ + F := n g ( k +1) k +1 , g ( k +1) k − o .In this way, we construct two sequences ( E , E , . . . , E k ) and ( F , F , . . . , F k )such that g ( k ) i ( y − y( k + 1)) E k +1 − i −−−−−→ + g ( k +1) i ,g ( k ) i Y ( i,k +1) ∈ S x (Ξ) ( x − x( σ xk +1 ( i ))) ! F k +1 − i −−−−−→ + , where, for i = 1 , . . . , k , E i = n g ( k +1) k +2 − i , . . . , g ( k +1) k +1 o ,F i = n g ( k +1) k +1 , g ( k +1) k +1 − i o , LM (cid:16) g ( k +1) i (cid:17) = x m i +1 y i . Let ¯ g ( k +1) i = LC (cid:16) g ( k +1) i (cid:17) − g ( k +1) i , i = 0 , . . . , k + 2. With similar argumentsused in N = 2 case, we can finally prove that G k +1 = n ¯ g ( k +1)0 , ¯ g ( k +1)1 , . . . , ¯ g ( k +1) k +2 o , with LM (cid:16) ¯ g ( k +1) i (cid:17) = x m i +1 y i , i = 0 , . . . , k +1, and LM (cid:16) ¯ g ( k +1) k +2 (cid:17) = y k +2 , namely(18) holds for N = k + 1. Consequently, we haveLM( G ν ) = { x m +1 , x m +1 y, . . . , x m ν +1 y ν , y ν +1 } . N ( I (Ξ)) = { x i y j : ( i, j ) ∈ S x (Ξ) } = M S x (Ξ) , which complete the proof. (cid:3) The ≺− degree of a nonzero polynomial f ∈ Π (see [3]), denoted by δ ≺ ( f ), wasdefined to be ( i, j ) ∈ N satisfying x i y j = LM ≺ ( f ) . For every pair of polynomials f, g ∈ Π , if δ ( f ) ≺ δ ( g ) then we say that f is of lower ≺− degree than g and use the abbreviation f ≺ g := δ ( f ) ≺ δ ( g ) . In addition, f (cid:22) g is interpreted as the ≺− degree of f is lower than or equal tothat of g .Given a finite nonempty set Ξ = { ξ (0) , ξ (1) , . . . , ξ ( µ − } ⊂ F . For fixedmonomial order ≺ , the Gr¨obner ´escalier N ≺ ( I (Ξ)) trivially forms the monomialbasis for µ -dimensional F -linear space P ≺ (Ξ) := Span F N ≺ ( I (Ξ)) ⊂ Π thatcomplements I (Ξ), i.e. Π = I (Ξ) ⊕ P ≺ (Ξ) . Moreover, if subset { p , p , . . . , p µ − } ⊂ P ≺ (Ξ), with p ≺ p ≺ · · · ≺ p µ − ,satisfying p j ( ξ ( i ) ) = δ ij , ≤ i ≤ j ≤ µ − , then { p , p , . . . , p µ − } is called a Newton interpolation basis for P ≺ (Ξ).Consequently, from Theorem 3.1, Ξ is x -tower (resp. y -tower) implies that P ≺ grlex (Ξ) = Span F M S x (Ξ) (resp. P ≺ grevlex (Ξ) = Span F M S y (Ξ) ). The next twotheorems present Newton bases for P ≺ grlex (Ξ) and P ≺ grevlex (Ξ) respectively. Theorem 3.2
Given an x -tower set Ξ ⊂ F that is expressed as (16) . Setpolynomial φ xij := c xij j − Y t =0 ( y − y( t )) i − Y s =0 ( x − x( σ xj ( s ))) , ( i, j ) ∈ S x (Ξ) , where c xij = 1 Q j − t =0 (y( j ) − y( t )) Q i − s =0 (x( σ xj ( i )) − x( σ xj ( s ))) ∈ F and the empty products are taken as 1. Then for ( i, j ) , ( m, n ) ∈ S x (Ξ) with ( i, j ) (cid:23) inlex ( m, n ) , we have φ xij (x( σ xn ( m )) , y( n )) = δ ( i,j ) , ( m,n ) , namely N S x (Ξ) := { φ xij : ( i, j ) ∈ S x (Ξ) } (19) forms a Newton interpolation basis for P ≺ grlex (Ξ) . roof: Fix ( i, j ) ∈ S x (Ξ). If ( i, j ) = ( m, n ), then φ xij (x( σ xn ( m )) , y( n )) = c xij j − Y t =0 (y( j ) − y( t )) i − Y s =0 (x( σ xj ( i )) − x( σ xj ( s ))) = c xij /c xij = 1 . Otherwise, ( i, j ) (cid:23) inlex ( m, n ) implies j > n or j = n and i > m . When j > n ,i.e., j − ≥ n , we have φ xij (x( σ xn ( m )) , y( n )) = c xij j − Y t =0 (y( n ) − y( t )) i − Y s =0 (x( σ xn ( m )) − x( σ xj ( s ))) = 0 . If j = n, i > m , namely i − ≥ m , then φ xij (x( σ xn ( m )) , y( n )) = c xij j − Y t =0 (y( n ) − y( t )) i − Y s =0 (x( σ xn ( m )) − x( σ xj ( s )))= c xij j − Y t =0 (y( n ) − y( t )) i − Y s =0 (x( σ xj ( m )) − x( σ xj ( s )))= 0 , which leads to φ xij (x( σ xn ( m )) , y( n )) = 0 , ( i, j ) ≻ inlex ( m, n ) , as desired. It is easy to check that Span F N x = Span F M S x (Ξ) = P ≺ grlex (Ξ). (cid:3) Similarly, we can prove the following theorem:
Theorem 3.3
Let Ξ ⊂ F be a y -tower set that is defined by (17) . We letpolynomial φ yij := c yij i − Y s =0 ( x − x( s )) j − Y t =0 ( y − y( σ yi ( t ))) , ( i, j ) ∈ S y (Ξ) , where c yij = 1 Q i − s =0 (x( i ) − x( s )) Q j − t =0 (y( σ yi ( j )) − y( σ yi ( t ))) ∈ F and the empty products are taken as 1. Then, N S y (Ξ) := { φ yij : ( i, j ) ∈ S y (Ξ) } (20) is a Newton interpolation basis for P ≺ grevlex (Ξ) satisfying φ yij ((x( m ) , y( σ ym ( n )))) = δ ( i,j ) , ( m,n ) , ( i, j ) (cid:23) lex ( m, n ) . Now, we turn to ≺ lex and ≺ inlex cases. For every finite nonempty set Ξ ⊂ F , [15] presents monomial and Newton interpolation bases for P ≺ lex (Ξ) and P ≺ inlex (Ξ). In the following, we restate the results with Ξ limited to tower setsonly. 16 emma 3.2 Let Ξ be an x -tower ( resp. y -tower ) set which is defined by (16)( resp. (17)) . Then M S x (Ξ) ( resp. M S y (Ξ) ) is the monomial basis and N S x (Ξ) ( resp. N S y (Ξ) ) a Newton basis for P ≺ lex (Ξ) ( resp. P ≺ inlex (Ξ)) . Combining Theorem 3.1–3.3 and Lemma 3.2, we arrive at the following maintheorem:
Theorem 3.4
Let Ξ be an x -tower ( resp. y -tower ) set that is defined by (16)( resp. (17)) . Then M S x (Ξ) ( resp. M S y (Ξ) ) is the monomial basis and N S x (Ξ) ( resp. N S y (Ξ) ) a Newton interpolation basis for P ≺ grlex (Ξ) and P ≺ lex (Ξ) ( resp. P ≺ grevlex (Ξ) and P ≺ inlex (Ξ)) . Now, it’s time for our improvement for Farr-Gao algorithm.
Algorithm 3.1
Input : A finite set Ξ ⊂ F and a fixed monomial order ≺ thatis either ≺ grlex or ≺ lex ( resp. either ≺ grevlex or ≺ inlex ) . Output : The reduced Gr¨obner basis for I (Ξ) w.r.t. ≺ . Step1.
Decompose Ξ following (7) and find an x -tower ( resp. y -tower ) subset Ξ T of Ξ as large as possible. Step2.
Obtain S x (Ξ T ) ( resp. S y (Ξ T )) following (8) , and finally express Ξ T in form (16) ( resp. (17)) . Step3.
Construct list P whose h -th entry p h , ≤ h ≤ T , is the h -thsmallest element of Ξ T in form (16) ( resp. (17)) w.r.t. increasing ≺ inlex ( resp. ≺ lex ) on ( i, j ) . Step4.
Compute set M := M S x (Ξ T ) ( resp. M S y (Ξ T ) ) and then set C := C [ M ] by applying (15) and (2) respectively. Step5.
Construct list N whose k -th entry q k , ≤ k ≤ T , is the k -th smallest element of N S x (Ξ T ) ( resp. N S y (Ξ T ) ) w.r.t. increasing ≺ inlex ( resp. ≺ lex ) on ( i, j ) . Step6.
Use
C, N to obtain the reduced Gr¨obner basis G for I (Ξ T ) . Step7.
Send G to Farr-Gao process to finish the computation. Algorithm 3.1 has been implemented on
Maple r
16 that is installed ona laptop with 8 Gb RAM and 2.3 GHz CPU. For ≺ grevlex , its running timeson 250, 500, and 1000 points in F q are compared with the build-in command VanishingIdeal of Maple r .When q = 41, we have ❳❳❳❳❳❳❳❳❳❳ Algorithms
VanishingIdeal q = 101, we have 17 ❳❳❳❳❳❳❳❳❳ Algorithms
VanishingIdeal
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