A divide-and-conquer algorithm for computing Gröbner bases of syzygies in finite dimension
AA Divide-and-conquer Algorithm for Computing Gröbner Basesof Syzygies in Finite Dimension
Simone Naldi
Univ. Limoges, CNRS, XLIM, UMR 7252
F-87000 Limoges, France
Vincent Neiger
Univ. Limoges, CNRS, XLIM, UMR 7252
F-87000 Limoges, France
ABSTRACT
Let f , . . . , f m be elements in a quotient R n /N which has finitedimension as a K -vector space, where R = K [ X , . . . , X r ] and N is an R -submodule of R n . We address the problem of computing aGröbner basis of the module of syzygies of ( f , . . . , f m ) , that is, ofvectors ( p , . . . , p m ) ∈ R m such that p f + · · · + p m f m = R n /N asthe kernel of a collection of linear functionals. Following this view-point, we design a divide-and-conquer algorithm, which can beinterpreted as a generalization to several variables of Beckermannand Labahn’s recursive approach for matrix Padé and rational in-terpolation problems. To highlight the interest of this method, wefocus on the specific case of bivariate Padé approximation and showthat it improves upon the best known complexity bounds. KEYWORDS
Syzygies; Gröbner basis; Padé approximation; divide and conquer
ACM Reference Format:
Simone Naldi and Vincent Neiger. 2020. A Divide-and-conquer Algorithmfor Computing Gröbner Bases of Syzygies in Finite Dimension. In
Inter-national Symposium on Symbolic and Algebraic Computation (ISSAC ’20),July 20–23, 2020, Athens, Greece.
ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/3373207.3404059
Context.
Hereafter, R = K [ X , . . . , X r ] is the ring of r -variatepolynomials over a field K . Given an R -submodule N ⊂ R n suchthat R n /N has finite dimension D as a K -vector space, as well as amatrix F ∈ R m × n with rows f , . . . , f m ∈ R n , this paper studiesthe computation of a Gröbner basis of the module of syzygiesSyz N ( F ) = { p = ( p i ) ≤ i ≤ m ∈ R m | pF = (cid:205) ≤ i ≤ m p i f i ∈ N } , where p is seen as a 1 × m row vector. Note that R m / Syz N ( F ) alsohas finite dimension, at most D , as a K -vector space.Following a path of work pioneered by Marinari, Möller andMora [1, 25, 27], we focus on a specific situation where N is de-scribed using duality. That is, N is known through D linear function-als φ j : R n → K such that N = ∩ ≤ j ≤ D ker ( φ j ) . In this context, itis customary to make an assumption equivalent to the following: N i = ∩ ≤ j ≤ i ker ( φ i ) is an R -module, for 1 ≤ i ≤ D ; see e.g. [25, ISSAC ’20, July 20–23, 2020, Athens, Greece © 2020 Association for Computing Machinery.This is the author’s version of the work. It is posted here for your personal use. Notfor redistribution. The definitive Version of Record was published in
InternationalSymposium on Symbolic and Algebraic Computation (ISSAC ’20), July 20–23, 2020,Athens, Greece , https://doi.org/10.1145/3373207.3404059.
Algo. 2] [16, Eqn. (4.1)] [30, Eqn. (5)] for such assumptions and re-lated algorithms. Namely, this assumption allows one to designiterative algorithms which compute bases of Syz N i ( F ) iterativelyfor increasing i , until reaching i = D and obtaining the sought basisof Syz N ( F ) . An efficient such iterative procedure is given in [25],specifically in Algorithm 2 (variant in Section 9 therein); note that itis written for m = n = F = [ ] , in which case Syz N i ( F ) = N i ,but directly extends to the case m ≥ F ∈ R m × n . Ideal of points and Padé approximation.
One particular case ofinterest is when N is the vanishing ideal of a given set of points: n =
1, and N is the ideal of all polynomials in R which vanishat distinct points α , . . . , α D ∈ K r . Here, one takes the linearfunctionals for evaluation: φ j : f ∈ R (cid:55)→ f ( α j ) ∈ K . The questionis, given the points, m polynomials as F ∈ R m × , and a monomialorder ≼ , to compute a ≼ -Gröbner basis of the set of vectors p suchthat pF vanishes at all the points. When m = F = [ ] , thismeans computing a ≼ -Gröbner basis of the ideal of the points, asstudied in [25, 26].Another case is that of (multivariate) Padé approximation andits extensions, as studied in [14, 16, 17, 30], as well as in [6] in thecontext of the computation of multidimensional linear recurrencerelations. The basic setting is for n =
1, with N an ideal of theform ⟨ X d , . . . , X d r r ⟩ , and F = [ f − ] for some given f ∈ R . Then,elements of Syz N ( F ) are vectors ( q , p ) ∈ R such that f = p / q mod X d , . . . , X d r r . Here, the D = d · · · d r linear functionals correspondto the coefficients of multidegree less than ( d , . . . , d r ) ; note thatnot all orderings of these functionals satisfy the assumption above.For these two situations, as well as some extensions of them, thefastest known algorithms rely on linear algebra and have a costbound of O ( mD + rD ) operations in K [16, 25]; this was recentlyimproved in [28, Thm. 2.13] and [29] to O ( mD ω − + rD ω log ( D )) where ω < .
38 is the exponent of matrix multiplication [10, 24].Based on work in [9, 15], in the specific case of an ideal of points N and the lexicographic order, Ceria and Mora gave a combina-torial algorithm to compute the ≼ lex -monomial basis of R/N , theCerlienco-Mureddu correspondence, and squarefree separators forthe points using O ( rD log ( D )) operations [8]. The univariate case.
This problem has received attention in thecase of a single variable ( r =
1) notably thanks to the numerousapplications of matrix rational interpolation and Hermite-Padé ap-proximation, which are the two situations described above. Iterativealgorithms were first given for Padé approximation in [18, 34] andthen for Hermite-Padé approximation in [2, 4, 33]; the latter can beseen as univariate analogues of [25, Algo. 2] and [16, Algo. 4.7].A breakthrough divide and conquer approach was designed byBeckermann and Labahn in [3, Algo. SPHPS], allowing one to take a r X i v : . [ c s . S C ] J un SSAC ’20, July 20–23, 2020, Athens, Greece Simone Naldi and Vincent Neiger advantage of univariate polynomial matrix multiplication whileprevious iterative algorithms only relied on naive linear algebraoperations. This led to a line of work [19, 21, 22, 32, 35] whichconsistently improved the incorporation of fast linear algebra andfast polynomial multiplication in this divide and conquer frame-work, culminating in cost bounds for rational interpolation andHermite-Padé approximation which are close asymptotically to thesize of the problem (if ω =
2, these cost bounds are quasi-linearin the size of the input). To the best of our knowledge, no similardivide and conquer technique has been developed in multivariatesettings prior to this work.
Contribution.
We propose a divide and conquer algorithm for theproblem of computing a ≼ -Gröbner basis of Syz N ( F ) in the multi-variate case. This is based on the iterative algorithm [25, Algo. 2],observing that each step of the iteration can be interpreted as a leftmultiplication by a matrix which has a specific shape, which wecall elementary Gröbner basis (see Section 3). The new algorithmreorganizes these matrix products through a divide and conquerstrategy, and thus groups several products by elementary Gröbnerbases into a single multivariate polynomial matrix multiplication.Thus, both the existing iterative and the new divide and conquerapproaches compute the same elementary Gröbner bases, but unlikethe former, our algorithm does not explicitly compute Gröbnerbases for all intermediate syzygy modules Syz N i ( F ) . By computingless, we expect to achieve better computational complexity. Toillustrate this, we specialize our approach to multivariate matrixPadé approximation and derive complexity bounds for this case; weobtain the next result, which is a particular case of Proposition 5.5.Theorem 1.1. For R = K [ X , Y ] , let f , . . . , f m ∈ R , and let ≼ bea monomial order on R . Then one can compute a minimal ≼ -Gröbnerbasis of the module of Hermite-Padé approximants {( p , . . . , p m ) ∈ R m | p f + · · · + p m f m = ⟨ X d , Y d ⟩} using O ˜ ( m ω d ω + ) operations in K , where O ˜ (·) means that polylog-arithmic factors are omitted. In this case the vector space dimension is D = d . Thus, as notedabove and to the best of our knowledge, the fastest previouslyknown algorithm for this task has a cost of O ˜ ( md ( ω − ) + d ω ) operations in K and does not exploit fast polynomial multiplication. Perspectives.
The base case of our divide and conquer algorithmconcerns the case N = ker ( φ ) of a single linear functional, detailedin Section 3; we thus work in a vector space R n /N of dimension 1.A natural perspective is to improve the efficiency of our algorithmthanks to a better exploitation of fast linear algebra by groupingseveral base cases together; using fast linear algebra to acceleratethe base case was a key strategy in obtaining efficient univariatealgorithms [19, 22]. In the context of Padé approximation, whereone can introduce the variables one after another, one could alsotry to incorporate known algorithms for the univariate case.One reason why these improvements are not straightforward todo in the multivariate case is that there is no direct generalizationof a property at the core of the correctness of univariate algorithms.This property (see [23, Lem. 2.4]) states that if P is a ≼ -Gröbnerbasis of N ⊃ N and P is a ≼ -Gröbner basis of Syz N ( P ) , then P P is a ≼ -Gröbner basis of N , provided that the order ≼ is well chosen (a Schreyer order for P and ≼ , see Section 2.4). We givea counterexample to such a property in Example 3.6. It remainsopen to find a similar general property that would help to designalgorithms based on matrix multiplication in the multivariate case.Another difficulty arises in analyzing the complexity of our di-vide and conquer scheme in contexts where the number of elementsin the sought Gröbner basis is not well controlled, such as rationalinterpolation. Indeed, this number corresponds to the size of thematrices used in the algorithm, and therefore is directly related tothe cost of the matrix multiplication. In fact, the worst-case numberof elements depends on the monomial order and is often pessimisticcompared to what is observed in a generic situation. Thus, futurework involves investigating complexity bounds for generic inputand for interesting particular cases other than Padé approximation. Here and hereafter, the coordinate vector with 1 at index i is denotedby e i ; its dimension is inferred from the context. A monomial in R m is an element of the form ν e i for some 1 ≤ i ≤ m and somemonomial ν in R ; i is called the support of ν e i . We denote byMon (R m ) the set of all monomials in R m . A term is a monomialmultiplied by a nonzero constant from K . The elements of R m are K -linear combinations of elements of Mon (R m ) and are calledpolynomials.Elements in R are written in regular font (e.g. monomials µ and ν and polynomials f and p ), while elements in R m are boldfaced(e.g. monomials µ and ν and polynomials f and p ). Vectors or(ordered) lists of polynomials in R m are seen as matrices, writtenin boldfaced capital letters; precisely, ( p , . . . , p k ) ∈ (R m ) k is seenas a matrix P ∈ R k × m whose i th row is p i . In particular, in whatfollows the default orientation is to see an element of R m as a row vector in R × m .For the sake of completeness, we recall below in Sections 2.2 to 2.4some classical definitions from commutative algebra concerningsubmodules of R m ; we assume familiarity with the correspondingnotions concerning ideals of R . For a more detailed introductionthe reader may refer to [11–13]. A monomial order on R m is a total order ≼ on Mon (R m ) suchthat, for ν ∈ Mon (R) and µ , µ ∈ Mon (R m ) with µ ≼ µ , onehas µ ≼ ν µ ≼ ν µ ; hereafter µ ≺ µ means that µ ≼ µ and µ (cid:44) µ . For p ∈ R m , its ≼ -leading monomial is denoted by lm ≼ ( p ) and is the largest of its monomials with respect to the order ≼ (wetake the convention lm ≼ ( ) = for ∈ R m the zero element). Weextend this notation to collections of polynomials P ⊂ R m withlm ≼ (P) = { lm ≼ ( p ) : p ∈ P} , and to matrices P ∈ R k × m withlm ≼ ( P ) the k × m matrix whose i th row is the ≼ -leading monomialof the i th row of P . Example 2.1.
The usual lexicographic comparison is a monomialorder on K [ X , Y ] : X a Y b ≼ lex X a ′ Y b ′ if and only if a < a ′ or ( a = a ′ and b < b ′ ). It can be used to define a monomial order on K [ X , Y ] , called the term-over-position lexicographic order: for Divide-and-conquer Algorithm for Computing Gröbner Bases of Syzygies in Finite Dimension ISSAC ’20, July 20–23, 2020, Athens, Greece µ , ν in Mon ( K [ X , Y ]) and i , j in { , } , µ e i ≼ toplex ν e j if and only if µ ≼ lex ν or ( µ = ν and i < j ).We refer to [11, Sec. 1.§2 and 5.§2] for other classical monomialorders, such as the degree reverse lexicographical order on R , andthe construction of term-over-position and position-over-term or-ders on R m from monomial orders on R .A monomial order ≼ on R m induces a monomial order ≼ i on R for each 1 ≤ i ≤ m , by restricting to the i th coordinate: for ν , ν ∈ Mon (R) , ν ≼ i ν if and only if ν e i ≼ ν e i . In particular,lm ≼ ( q p ) is a multiple of lm ≼ ( p ) for q ∈ R and p ∈ R m :Lemma 2.2. Let ¯ ı be the support of lm ≼ ( p ) . Then lm ≼ ( q p ) = lm ≼ ¯ ı ( q ) lm ≼ ( p ) . Proof. Write q = (cid:205) ℓ ν ℓ and p = (cid:205) i , j µ ij e i for terms µ ij , ν ℓ in R . Then q p = (cid:205) ℓ, i , j ν ℓ µ ij e i , i.e. the terms of q p are all those ofthe form ν ℓ µ ij e i . Now let ¯ ℓ and ¯ ȷ be such that lm ≼ ¯ ı ( q ) = ν ¯ ℓ andlm ≼ ( p ) = µ ¯ ı ¯ ȷ e ¯ ı . Then ν ℓ ≺ ¯ ı ν ¯ ℓ for all ℓ (cid:44) ¯ ℓ , which implies that ν ℓ µ ¯ ı ¯ ȷ ≺ ¯ ı ν ¯ ℓ µ ¯ ı ¯ ȷ and thus, by definition of ≼ ¯ ı , that ν ℓ µ ¯ ı ¯ ȷ e ¯ ı ≺ ν ¯ ℓ µ ¯ ı ¯ ȷ e ¯ ı .On the other hand, µ ij e i ≺ µ ¯ ı ¯ ȷ e ¯ ı holds for all ( i , j ) (cid:44) ( ¯ ı , ¯ ȷ ) , hence ν ℓ µ ij e i ≺ ν ℓ µ ¯ ı ¯ ȷ e ¯ ı . Therefore we obtain ν ℓ µ ij e i ≼ ν ¯ ℓ µ ¯ ı ¯ ȷ e ¯ ı for all ( i , j , ℓ ) , with equality only if ( i , j , ℓ ) = ( ¯ ı , ¯ ȷ , ¯ ℓ ) . This proves thatlm ≼ ( q p ) = ν ¯ ℓ µ ¯ ı ¯ ȷ e ¯ ı = lm ≼ ¯ ı ( q ) lm ≼ ( p ) . □ As a consequence of Hilbert’s Basis Theorem, any R -submoduleof R m is finitely generated [13, Prop. 1.4]. For a (possibly infi-nite) collection of polynomials P ⊂ R m , we denote by ⟨P⟩ the R -submodule of R m generated by the elements of P . Similarly,for a matrix P in R k × m , ⟨ P ⟩ stands for the R -submodule of R m generated by its rows, that is, ⟨ P ⟩ = { qP | q ∈ R k } .For a given submodule M ⊂ R m , the ≼ -leading module of M isthe module ⟨ lm ≼ (M)⟩ generated by the leading monomials of theelements of M . Then, a matrix P in R k × m whose rows are in M issaid to be a ≼ -Gröbner basis of M if ⟨ lm ≼ (M)⟩ = ⟨ lm ≼ ( P )⟩ . In this case we have ⟨ P ⟩ = M (see [11, Ch.5, Prop.2.7]), hence wewill often omit the reference to the module M and just say that P is a ≼ -Gröbner basis.A ≼ -Gröbner basis P , whose rows are ( p , . . . , p k ) , is said tobe minimal if lm ≼ ( p i ) is not divisible by lm ≼ ( p j ) , for any j (cid:44) i .It is said to be reduced if it is minimal and, for all 1 ≤ i ≤ k ,lm ≼ ( p i ) is monic and none of the terms of p i is divisible by any of { lm ≼ ( p j ) | j (cid:44) i } . Given a monomial order ≼ and an R -submodule M ⊂ R m , there is a reduced ≼ -Gröbner basis of M and it is unique(up to permutation of its elements) [13, Sec. 15.2]. Example 2.3.
The syzygy module M = {( p , p ) ∈ K [ X , Y ] | p − p ∈ ⟨ X , Y ⟩} = Syz ⟨ X , Y ⟩ ([ − ]) is generated by ( X e , Y e , e + e ) , that is, by the rows of P = X Y
01 1 ∈ K [ X , Y ] × . Furthermore, P is the reduced ≼ toplex -Gröbner basis of M . In the context of the computation of bases of syzygies it is generallybeneficial to use a specific construction of monomial orders, as firsthighlighted by Schreyer [20, 31] (see also [13, Th. 15.10] and [5]).In the univariate case, the notion of shifted degree plays the samerole as Schreyer orders and is ubiquitous in the computation of basesof modules of syzygies [19, 21, 35]; an equivalent notion of defectswas also used earlier for M-Padé and Hermite-Padé approximationalgorithms [2, 3]. Specifically, this provides a monomial order on R k constructed from a monomial order ≼ on R m and from theleading monomials of a ≼ -Gröbner basis in R m of cardinality k . Definition 2.4.
Let ≼ be a monomial order on R m , and let L = ( µ , . . . , µ k ) be a list of monomials of R m . A Schreyer order for ≼ and L is any monomial order on R k , denoted by ≼ L , such that for ν e i , ν e j ∈ Mon (R k ) , if ν µ i ≺ ν µ j then ν e i ≼ L ν e j .As noted above, this notion is often used with L = lm ≼ ( P ) for a listof polynomials P ∈ R k × m , which is typically a ≼ -Gröbner basis.Remark that Definition 2.4 uses a strict inequality, and impliesthat if ν e i ≼ L ν e j , then ν µ i ≺ ν µ j or ν µ i = ν µ j . In particular,for ν = ν = µ i (cid:44) µ j for all i (cid:44) j (for instance, if L = lm ≼ ( P ) for a minimal ≼ -Gröbner basis P ), then e i ≼ L e j if andonly if µ i ≺ µ j .Furthermore, for every ≼ and L , a corresponding Schreyer orderexists and can be constructed explicitly: for example, ν e i ≼ L ν e j if and only if ν µ i ≺ ν µ j or ( ν µ i = ν µ j and i < j ) . This specific Schreyer order is the one used in the algorithms inthis paper, where we write ≼ L ← SchreyerOrder ( ≼ , L ) to mean that the algorithm constructs it from ≼ and L . In this section we present the base case of our main algorithm. Itconstructs Gröbner bases for syzygies modulo the kernel of a singlelinear functional, which we call elementary Gröbner bases anddescribe in Section 3.1. Further in Section 3.2 we state propertiesthat are useful to prove the correctness of the base case algorithmgiven in Section 3.3. Precisely, this correctness is written havingin mind the design of an algorithm handling several functionalsiteratively by repeating this basic procedure and multiplying theelementary bases together. If I ⊂ R is an ideal such that
R/I has dimension 1 as a K -vectorspace, then I is maximal: it is of the form ⟨ X − α , . . . , X r − α r ⟩ for some point ( α , . . . , α r ) ∈ K r , which directly yields the reducedGröbner basis of I , for any monomial order. In this paper, wewill make use of a similar property for submodules of R m ; suchsubmodules have Gröbner bases of the form E = I π − λ X − αλ I m − π ∈ R ( m + r − )× m , (1) SSAC ’20, July 20–23, 2020, Athens, Greece Simone Naldi and Vincent Neiger for the vector of variables X = [ X · · · X r ] T and vectors of values α = [ α · · · α r ] T ∈ K r × , λ = [ λ · · · λ π − ] T ∈ K ( π − )× , and λ = [ λ π + · · · λ m ] T ∈ K ( m − π )× . In what follows, such matricesare called elementary Gröbner bases.Theorem 3.1. Let M be an R -submodule of R m such that R m /M has dimension as a K -vector space, then for any monomial order ≼ on R m , the reduced ≼ -Gröbner basis E of M is as in Eq. (1) with λ i = if e i ≺ e π for all i (cid:44) π . Conversely, any matrix E as in Eq. (1) defines a submodule M = ⟨ E ⟩ such that R m /M has dimension as a K -vector space, and E is a reduced ≼ -Gröbner basis for any monomialorder ≼ such that λ i = if e i ≺ e π for all i (cid:44) π . Proof. By [13, Thm. 15.3], a basis of R m /M as a K -vector spaceis given by the monomials not in lm ≼ (M) ; since the dimension of R m /M as a K -vector space is 1, there exists a unique monomialwhich is not in lm ≼ (M) . Thus there is a unique π ∈ { , . . . , m } such thatlm ≼ ( E ) = ( e , . . . , e π − , X e π , . . . , X r e π , e π + , . . . , e m ) . (2)By definition of reduced Gröbner bases, the j th polynomial in E isthe sum of the j th element of lm ≼ ( E ) and a constant multiple of e π ;hence E has the form in Eq. (1). In addition, for i (cid:44) π , the equalitylm ≼ ( e i + λ i e π ) = e i implies that λ i = e i ≺ e π .For the converse, let ≼ be such that λ i = e i ≺ e π for all i (cid:44) π (such an order exists since there are orders for which e π is the smallest coordinate vector). Then lm ≼ ( E ) is as in Eq. (2); inparticular, the monomials in ⟨ lm ≼ ( E )⟩ are precisely Mon (R m ) \{ e π } . It follows that either e π ∈ lm ≼ (M) and ⟨ lm ≼ (M)⟩ = R m ,or e π (cid:60) lm ≼ (M) and ⟨ lm ≼ (M)⟩ = ⟨ lm ≼ ( E )⟩ . In the second case E is a reduced ≼ -Gröbner-basis and R m /M has dimension 1 by[13, Thm. 15.3]. To conclude the proof, we show that e π ∈ lm ≼ (M) cannot occur; by contradiction, suppose there exists q ∈ M suchthat lm ≼ ( q ) = e π . Since the rows of E generate M , we can write q = ( q , . . . , q π − , p , . . . , p r , q π + , . . . , q m ) E = (cid:169)(cid:173)(cid:171) q , . . . , q π − , (cid:213) i (cid:44) π q i λ i + r (cid:213) j = ( X j − α j ) p j , q π + , . . . , q m (cid:170)(cid:174)(cid:172) . For i (cid:44) π such that e π ≺ e i , any nonzero term of q i e i wouldappear in q and be greater than e π , hence q i =
0. Moreover, for i (cid:44) π such that e i ≺ e π we have λ i =
0. Thus, considering the π thcomponent of q yields the equality1 = (cid:213) i (cid:44) π q i λ i + r (cid:213) j = ( X j − α j ) p j = r (cid:213) j = ( X j − α j ) p j which is a contradiction since 1 (cid:60) ⟨ X − α , . . . , X r − α r ⟩ . □ Remark that in the module case ( m ≥
2) the reduced ≼ -Gröbnerbasis depends on the order ≼ , more precisely on how the e i ’s areordered by ≼ . For instance, the matrix in Example 2.3 is a reduced ≼ -Gröbner basis for every order such that e ≼ e , whereas fororders such that e ≼ e the reduced ≼ -Gröbner basis of the samemodule is E = X Y ∈ K [ X , Y ] × . Let ≼ be a monomial order on R m and let P = ( p , . . . , p k ) ∈ R k × m be a ≼ -Gröbner basis. In this section, we show conditions on anelementary Gröbner basis E to ensure that EP is a ≼ -Gröbner basis.We write L = ( µ , . . . , µ k ) for lm ≼ ( P ) , that is, µ i = lm ≼ ( p i ) for1 ≤ i ≤ k . Let ≼ L be a Schreyer order for ≼ and P , and consider areduced ≼ L -Gröbner basis E ∈ R ( k + r − )× k which has the form inEq. (1); thus EP = ( p + λ p π , . . . , p π − + λ π − p π , ( X − α ) p π , . . . , ( X r − α r ) p π , λ π + p π + p π + , . . . , λ k p π + p k ) which is in R ( k + r − )× m . We will show that, under suitable assump-tions, EP is a ≼ -Gröbner basis; the next lemmas use the abovenotation. We start by describing the leading terms of EP .Lemma 3.2. If µ i (cid:44) µ π for all i (cid:44) π , then lm ≼ ( EP ) = lm ≼ L ( E ) L = ( µ , . . . , µ π − , X µ π , . . . , X r µ π , µ π + , . . . , µ k ) . Proof. First, lm ≼ (( X j − α j ) p π ) = X j µ π for 1 ≤ j ≤ r . Next weclaim that lm ≼ ( p i + λ i p π ) = µ i for all i (cid:44) π . If λ i =
0, the identityis obvious. If λ i (cid:44)
0, then e π ≼ L e i (see Section 3.1), and fromthe definition of a Schreyer order and the assumption µ π (cid:44) µ i , wededuce µ π ≺ µ i and hence lm ≼ ( p i + λ i p π ) = µ i . □ Next, we characterize the fact that EP generates a submodulewhich differs from the one generated by P .Lemma 3.3. If µ i (cid:44) µ π for all i (cid:44) π , then ⟨ EP ⟩ (cid:44) ⟨ P ⟩ ⇔ p π (cid:60) ⟨ EP ⟩ ⇔ µ π (cid:60) ⟨ lm ≼ (⟨ EP ⟩)⟩ . Proof. First, remark that ⟨ EP ⟩ = ⟨ P ⟩ ⇒ p π ∈ ⟨ EP ⟩ ⇒ µ π ∈⟨ lm ≼ (⟨ EP ⟩)⟩ is obvious; thus, to conclude the proof it remainsto show that ⟨ EP ⟩ = ⟨ P ⟩ ⇐ µ π ∈ ⟨ lm ≼ (⟨ EP ⟩)⟩ . Suppose that µ π ∈ ⟨ lm ≼ (⟨ EP ⟩)⟩ . Then, since µ i ∈ ⟨ lm ≼ (⟨ EP ⟩)⟩ for all i (cid:44) π by Lemma 3.2, we have lm ≼ ( P ) ⊂ ⟨ lm ≼ (⟨ EP ⟩)⟩ , hence ⟨ lm ≼ ( P )⟩ ⊂⟨ lm ≼ (⟨ EP ⟩)⟩ . Furthermore, recall that ⟨ lm ≼ ( P )⟩ = ⟨ lm ≼ (⟨ P ⟩)⟩ since P is a ≼ -Gröbner basis, and that ⟨ lm ≼ (⟨ EP ⟩)⟩ ⊂ ⟨ lm ≼ (⟨ P ⟩)⟩ since ⟨ EP ⟩ ⊂ ⟨ P ⟩ : we obtain ⟨ lm ≼ (⟨ P ⟩)⟩ = ⟨ lm ≼ (⟨ EP ⟩)⟩ . Then, [13,Lemma 15.5] shows that ⟨ EP ⟩ = ⟨ P ⟩ . □ For example, if P is a minimal ≼ -Gröbner basis, then the assump-tion in the previous lemma is satisfied. Example 3.6 below exhibitsa case where P is a minimal ≼ -Gröbner basis and p π does belong to ⟨ EP ⟩ . In that case, ⟨ EP ⟩ = ⟨ P ⟩ and EP is not a Gröbner basis since µ π is in ⟨ lm ≼ (⟨ EP ⟩)⟩ but not in ⟨ lm ≼ ( EP )⟩ .Lemma 3.4. If µ i (cid:44) µ π for all i (cid:44) π and ⟨ EP ⟩ (cid:44) ⟨ P ⟩ , then EP isa ≼ -Gröbner basis. Proof. Suppose by contradiction that EP is not a ≼ -Gröbnerbasis. Then there exists a nonzero h ∈ ⟨ EP ⟩ such that lm ≼ ( h ) (cid:60) ⟨ lm ≼ ( EP )⟩ , that is, by Lemma 3.2, lm ≼ ( h ) is not divisible by any ofthe elements µ i for i (cid:44) π and X j µ π for 1 ≤ j ≤ r . On the otherhand, lm ≼ ( h ) is in ⟨ lm ≼ (⟨ EP ⟩)⟩ and therefore in ⟨ lm ≼ ( P )⟩ , hencelm ≼ ( h ) is divisible by at least one µ i , 1 ≤ i ≤ k . These divisibilityconstraints lead to lm ≼ ( h ) = µ π , which implies µ π ∈ ⟨ lm ≼ (⟨ EP ⟩)⟩ .From Lemma 3.3 one deduces ⟨ EP ⟩ = ⟨ P ⟩ , which is absurd. □ Divide-and-conquer Algorithm for Computing Gröbner Bases of Syzygies in Finite Dimension ISSAC ’20, July 20–23, 2020, Athens, Greece
Corollary 3.5.
Assume that ⟨ EP ⟩ (cid:44) ⟨ P ⟩ and that P is a minimal ≼ -Gröbner basis. Let j < · · · < j ℓ be the indices j ∈ { , . . . , r } suchthat X j µ π (cid:60) ⟨ µ i , i (cid:44) π ⟩ . Then, the submatrix Q = I π − λ X j − α j ... X j ℓ − α j ℓ λ I m − π ∈ R ( k + ℓ − )× k (3) of E is such that QP is a minimal ≼ -Gröbner basis of ⟨ EP ⟩ . Proof. Since P is minimal, µ i (cid:44) µ π for all i (cid:44) π ; then Lemma 3.4ensures that EP is a ≼ -Gröbner basis and Lemma 3.2 giveslm ≼ ( QP ) = ( µ , . . . , µ π − , X j µ π , . . . , X j ℓ µ π , µ π + , . . . , µ k ) . By construction of j , . . . , j ℓ , one has ⟨ lm ≼ ( QP )⟩ = ⟨ lm ≼ ( EP )⟩ ,which implies ⟨ lm ≼ (⟨ EP ⟩)⟩ = ⟨ lm ≼ ( EP )⟩ = ⟨ lm ≼ ( QP )⟩⊂ ⟨ lm ≼ (⟨ QP ⟩)⟩ ⊂ ⟨ lm ≼ (⟨ EP ⟩)⟩ . Hence ⟨ lm ≼ ( QP )⟩ = ⟨ lm ≼ (⟨ QP ⟩)⟩ , and QP is a minimal ≼ -Gröbnerbasis. We conclude using [13, Lem. 15.5], which shows that ⟨ QP ⟩ ⊂⟨ EP ⟩ and ⟨ lm ≼ (⟨ QP ⟩)⟩ = ⟨ lm ≼ (⟨ EP ⟩)⟩ imply ⟨ QP ⟩ = ⟨ EP ⟩ . □ Example 3.6.
Consider the case R = K [ X , Y ] and m =
1. Let P = [ XY + ] ∈ R × , which is the reduced ≼ -Gröbner basis of ⟨ X , Y + ⟩ for any monomial order ≼ on Mon (R) . Let also E ∈ R × whose rows are ( X e , Y e , e ) ; according to Theorem 3.1, E is areduced ≼ -Gröbner basis for any monomial order ≼ on Mon (R ) .Now, the product EP ∈ R × has entries X , XY , and Y +
1. Thus, ⟨ lm ≼ ( EP )⟩ = ⟨ X , XY , Y ⟩ = ⟨ X , Y ⟩ for any monomial order ≼ on Mon (R) . On the other hand, ⟨ EP ⟩ contains X = X ( Y + ) − XY ,hence ⟨ lm ≼ ( EP )⟩ (cid:44) ⟨ lm ≼ (⟨ EP ⟩)⟩ , which means that EP is not a ≼ -Gröbner basis. We now describe Algorithm Syzygy_BaseCase, which will serveas the base case of the divide and conquer scheme.Theorem 3.7.
Let
N ⊂ R n be an R -submodule, let F ∈ R m × n ,and let P ∈ R k × m be a minimal ≼ -Gröbner basis of Syz N ( F ) for somemonomial order ≼ on R m . Assume that the input of Algorithm 1is such that ker ( φ ) ∩ N is an R -module, G = PF , and lm ≼ ( P ) = ( µ , . . . , µ k ) . Then Algorithm 1 returns ( Q , L ) such that QP is aminimal ≼ -Gröbner basis of Syz ker ( φ )∩N ( F ) and L = lm ≼ ( QP ) . Proof. If ( φ ( д ) , . . . , φ ( д k )) = ( , . . . , ) , then Algorithm 1 stopsat Line 2 and returns Q = I k and K . Thus QP = P , hence by assump-tion L = K = lm ≼ ( P ) = lm ≼ ( QP ) , and QP is a minimal ≼ -Gröbnerbasis of Syz N ( F ) ; besides, the identity Syz N ( F ) = Syz ker ( φ )∩N ( F ) is easily deduced from ( φ ( д ) , . . . , φ ( д k )) = ( , . . . , ) .In the rest of the proof, assume ( φ ( д ) , . . . , φ ( д k )) (cid:44) ( , . . . , ) .Define E ∈ R ( k + r − )× k as in Eq. (1) with π and λ i as in Algorithm 1and α j = φ ( X j д π )/ υ π for 1 ≤ j ≤ r ; in particular, Q computed atLine 8 is formed by a subset of the rows of E .First, E is a ≼ K -Gröbner basis according to Theorem 3.1, sinceby definition of π and λ i one gets the implications e i ≼ K e π ⇒ υ i = ⇒ λ i =
0, for i (cid:44) π . Algorithm 1
Syzygy_BaseCase ( φ , G , ≼ , L ) Input: • a linear functional φ : R n → K ,• a matrix G in R k × n with rows д , . . . , д k ∈ R n ,• a monomial order ≼ on R m ,• a list K = ( µ , . . . , µ k ) of elements of Mon (R m ) . Output: • a matrix Q in R ( k + ℓ − )× k for some ℓ ∈ { , . . . , r } ,• a list L of k + ℓ − (R k ) . ( υ , . . . , υ k ) ← ( φ ( д ) , . . . , φ ( д k )) ∈ K k if ( υ , . . . , υ k ) = ( , . . . , ) then return ( I k , K ) ≼ K ← SchreyerOrder ( ≼ , K ) π ← arg min ≼ K { e i | ≤ i ≤ k , υ i (cid:44) } ▷ the index i such that υ i (cid:44) which minimizes e i with respect to ≼ K { j < · · · < j ℓ } ← { j ∈ { , . . . , r } | X j µ π (cid:60) ⟨ µ i , i (cid:44) π ⟩} α j s ← φ ( X j s д π )/ υ π for 1 ≤ s ≤ ℓ λ i ← − υ i / υ π for 1 ≤ i < π and π < i ≤ k Q ← matrix in R ( k + ℓ − )× k as in Eq. (3) L ← ( µ , . . . , µ π − , X j µ π , . . . , X j ℓ µ π , µ π + , . . . , µ k ) return ( Q , L ) Next, we claim that ⟨ E ⟩ = Syz ker ( φ )∩N ( G ) . Indeed, the rows of PF are in N , and thus so are those of EG = EPF . Moreover, bychoice of π and λ i the rows of EG are in ker ( φ ) , since for i (cid:44) π one has φ (( p i + λ i p π ) F ) = φ ( д i + λ i д π ) = υ i + λ i υ π = ≤ j ≤ r one has φ (( X j − α j ) p π F ) = φ (( X j − α j ) д π ) = φ ( X j д π ) − α j υ π =
0. Therefore the rows of EG are in ker ( φ ) ∩ N ,that is, ⟨ E ⟩ ⊂ Syz ker ( φ )∩N ( G ) . To prove the reverse inclusion, re-call from Theorem 3.1 that ⟨ E ⟩ has codimension 1 in R k and henceSyz ker ( φ )∩N ( G ) is either ⟨ E ⟩ or R k . Since0 (cid:44) υ π = φ ( д π ) = φ ( p π F ) = φ ( e π PF ) = φ ( e π G ) one has that e π (cid:60) Syz ker ( φ )∩N ( G ) , hence Syz ker ( φ )∩N ( G ) = ⟨ E ⟩ .It follows that ⟨ EP ⟩ = Syz ker ( φ )∩N ( F ) . Indeed, the rows of EPF are in ker ( φ ) ∩ N as noted above, and thus ⟨ EP ⟩ ⊂ Syz ker ( φ )∩N ( F ) .Now let p ∈ Syz ker ( φ )∩N ( F ) ; thus in particular p ∈ Syz N ( F ) , and p = qP for some q ∈ R k . Then pF = qPF = qG ∈ ker ( φ ) ∩ N ,hence q ∈ Syz ker ( φ )∩N ( G ) = ⟨ E ⟩ , and therefore p ∈ ⟨ EP ⟩ .Now, φ ( p π F ) (cid:44) p π (cid:60) Syz ker ( φ )∩N ( F ) = ⟨ EP ⟩ . ThusLemma 3.3 ensures ⟨ EP ⟩ (cid:44) ⟨ P ⟩ , and finally Corollary 3.5 statesthat QP is a minimal ≼ -Gröbner basis of ⟨ EP ⟩ = Syz ker ( φ )∩N ( F ) .Besides Lemma 3.2 yields lm ≼ ( QP ) = lm ≼ K ( Q ) K = L . □ Repeating the basic procedure described in Section 3.3 iteratively,we obtain an algorithm for syzygy basis computation when N is an intersection of kernels of linear functionals with a specificproperty (see Eq. (4)). This algorithm is similar to [25, Algo. 2] and[30, Algo. 3.2], apart from differences in the input description. Here,the input consists of linear functionals φ , . . . , φ D : R n → K , withthe assumption that N i = ∩ ≤ j ≤ i ker ( φ j ) is an R -module for 1 ≤ i ≤ D . (4)Then we consider the R -module N = N D = ∩ ≤ j ≤ D ker ( φ j ) ,which is such that R n /N has dimension at most D as a K -vector SSAC ’20, July 20–23, 2020, Athens, Greece Simone Naldi and Vincent Neiger space. For F in R m × n , the following algorithm computes a minimal ≼ -Gröbner basis of the syzygy module Syz N ( F ) . Note that we donot specify the representation of F since it may depend on the spe-cific functionals φ i ; typically, one considers F to be known modulo N , via the images of its rows by the functionals φ i . Algorithm 2
Syzygy_Iter ( φ , . . . , φ D , F , ≼ ) Input: • linear functionals φ , . . . , φ D : R n → K such that Eq. (4),• a matrix F in R m × n ,• a monomial order ≼ on R m . Output: • a minimal ≼ -Gröbner basis P ∈ R k × m of Syz N ( F ) . P ← I m ∈ R m × m ; G ← F ; L ← ( e , . . . , e m ) = lm ≼ ( P ) for i = , . . . , D do ( Q , L ) ← Syzygy_BaseCase ( φ i , G , ≼ , L ) P ← QP ; G ← QG return P Corollary 4.1.
At the end of the i th iteration of Algorithm 2, P is a minimal ≼ -Gröbner basis of Syz N i ( F ) , and one has G = PF aswell as L = lm ≼ ( P ) . In particular, Algorithm 2 is correct. Proof. Note that at Line 1 of Algorithm 2, P = I m is the reduced ≼ -Gröbner basis of R m = Syz N ( F ) with N = R n , and both G = PF = F and L = ( e , . . . , e m ) = lm ≼ ( P ) hold. We concludethat if D =
0, Algorithm 2 is correct.The rest of the proof is by induction on D . We claim that theproperties in the statement are preserved across the D iterations.Precisely, we assume that at the beginning of the i th iteration, P isa minimal ≼ -Gröbner basis of Syz N i ( F ) , G = PF , and L = lm ≼ ( P ) .Since N i + = ker ( φ i + ) ∩ N i is an R -module, applying Theo-rem 3.7 shows that ( Q , L ) computed during the iteration are suchthat L = lm ≼ ( QP ) and that QP is a minimal ≼ -Gröbner basis ofSyz N i + ( F ) . □ This allows us to deduce bounds on the size of a minimal ≼ -Gröbner basis of Syz N ( F ) .Lemma 4.2. Let P ∈ R k × m be the output of Algorithm 2. Then, m ≤ k ≤ m + ( r − ) D , and thus the same holds for any minimal ≼ -Gröbner basis of Syz N ( F ) . Furthermore, at the end of the iteration i of Algorithm 2, the basis Q has at most k + D − i elements. Proof. Remark that all minimal ≼ -Gröbner bases of the samemodule have the same number of rows. Before the first iteration,the basis is I m which has m rows, and each iteration of the forloop adds ℓ − ℓ in { , . . . , r } . Therefore k ≤ m + ( r − ) D , and the last claim follows from ℓ − ≥ −
1. Thelower bound m ≤ k comes from the fact that R m / Syz N ( F ) hasfinite dimension as a K -vector space. □ This iterative algorithm can be turned into a divide and conquerone (Algorithm 3), by reorganizing how the products are performed.It computes a minimal ≼ -Gröbner basis of Syz N ( F ) , if one takes asinput G = F and K = ( e , . . . , e m ) . Algorithm 3
Syzygy_DaC ( φ , . . . , φ D , G , ≼ , K ) Input: • linear functionals φ , . . . , φ D : R n → K ,• a matrix G in R k × n ,• a monomial order ≼ on R m ,• a list K = ( µ , . . . , µ k ) of elements of Mon (R m ) . Output: • a matrix Q in R ℓ × m for some ℓ ≥ L of ℓ elements of Mon (R m ) . if D = then return Syzygy_BaseCase( φ i , G , ≼ , K ) ( Q , L ) ← Syzygy_DaC ( φ , . . . , φ ⌊ D / ⌋ , G , ≼ , K ) ( Q , L ) ← Syzygy_DaC ( φ ⌊ D / ⌋ + , . . . , φ D , Q G , ≼ , L ) return ( Q Q , L ) Theorem 4.3.
Let
N ⊂ R n be an R -submodule, let F ∈ R m × n ,and let P ∈ R k × m be a minimal ≼ -Gröbner basis of Syz N ( F ) for somemonomial order ≼ on R m . Assume that the input of Algorithm 3 issuch that G = PF , and lm ≼ ( P ) = ( µ , . . . , µ k ) , and N i ∩ N is an R -module for ≤ i ≤ D , (5) where N i = ∩ ≤ j ≤ i ker ( φ j ) . Then Algorithm 3 outputs ( Q , L ) suchthat QP is a minimal ≼ -Gröbner basis of Syz N D ∩N ( F ) and L = lm ≼ ( QP ) . Proof. If D = QP is a minimal ≼ -Gröbner basis ofSyz ker ( φ )∩N ( F ) and L = lm ≼ ( QP ) . We assume by induction hy-pothesis that Algorithm 3 returns the output foreseen by Theo-rem 4.3 when the number of input linear functionals is < D , andwhen the assumptions of the theorem are satisfied.By such a hypothesis, since G = PF and K = lm ≼ ( P ) , onededuces that ( Q , L ) are such that Q P is a ≼ -Gröbner basis ofSyz M ( F ) , with M = N ⌊ D / ⌋ ∩ N , and L = lm ≼ ( Q P ) .Let K i = ∩ ⌊ D / ⌋ + ≤ j ≤ i ker ( φ j ) , for each i = ⌊ D / ⌋ + , . . . , D .By hypothesis K i ∩ M = N i ∩ N is a module, for i = ⌊ D / ⌋ + i = D . Since Q G = Q PF and Q P is a ≼ -Gröbner basis ofSyz M ( F ) , and L = lm ≼ ( Q P ) , we can apply again the inductionhypothesis, and conclude that ( Q , L ) is such that Q Q P is aminimal ≼ -Gröbner basis of Syz K D ∩M ( F ) = Syz N D ∩N ( F ) , and L = lm ≼ ( Q Q P ) . We conclude that the global output ( Q Q , L ) satisfies the claimed properties. □ The algorithm in the previous section gives a general framework,which can be refined when applied to a particular context. Here,we consider the context of multivariate Padé approximation, where N = ⟨ X d , . . . , X d r r ⟩ × · · · × ⟨ X d , . . . , X d r r ⟩ ⊆ R n , (6)for some d , . . . , d r ∈ Z > . We begin with some remarks on thedegrees and sizes of Gröbner bases of syzygy modules Syz N ( F ) .To express this context in the framework of Section 4, we take forthe D linear functionals φ i the dual basis of the canonical monomialbasis of R n /N . Precisely, the linear functionals are φ µ , j : R n → K for 1 ≤ j ≤ n and all monomials µ ∈ Mon (R) with deg X i ( µ ) < d i for 1 ≤ i ≤ r , defined as follows: for f = ( f , . . . , f n ) ∈ R n , φ µ , j ( f ) is the coefficient of the monomial µ in f j . These linear functionals Divide-and-conquer Algorithm for Computing Gröbner Bases of Syzygies in Finite Dimension ISSAC ’20, July 20–23, 2020, Athens, Greece can be ordered in several ways to ensure that Eq. (4) is satisfied.Here we design our algorithm by ordering the functionals φ µ , j according to the term-over-position lexicographic order on themonomials µ e j ∈ Mon (R n ) . Example 5.1.
Consider the case of r = X , Y with d = d =
4, and n =
2. Then the functionals are φ , , φ , , φ Y , , φ Y , , φ Y , , φ Y , , φ Y , , φ Y , , φ X , , φ X , , φ XY , , φ XY , , φ XY , , φ XY , , φ XY , , φ XY , , in this specific order.Lemma 5.2. Let N be as in Eq. (6) , let F ∈ R m × n , and let ≼ be amonomial order on R m . Then, for ≤ i ≤ r , each polynomial in thereduced ≼ -Gröbner basis of Syz N ( F ) either has degree in X i less than d i or has the form X d i i e j for some ≤ j ≤ m . Proof. Let P be the reduced ≼ -Gröbner basis of Syz N ( F ) andlet i ∈ { , . . . , r } . Since R m / Syz N ( F ) has finite dimension as a K -vector space, for each j ∈ { , . . . , n } there is a polynomial in P whose ≼ -leading monomial has the form X di e j for some d ≥
0. Since P is reduced, any other ( p , . . . , p m ) in P whose ≼ -leading mono-mial has support j is such that deg X i ( p j ) < d ≤ d i ; the last inequal-ity follows from the fact that the monomial X d i i e j is in Syz N ( F ) and thus is a multiple of X di e j . It follows that all polynomials in P whose ≼ -leading monomial is not among { X d i i e j , ≤ j ≤ n } musthave degree in X i less than d i . On the other hand, any polynomialin P whose ≼ -leading monomial is X d i i e j for some j must be equalto this monomial, since it belongs to Syz N ( F ) and P is reduced. □ In the context of Algorithm 3, Lemma 5.2 allows us to truncatethe product Q Q while preserving a ≼ -Gröbner basis.Corollary 5.3. Let N be as in Eq. (6) , let F ∈ R m × n , let ≼ be a monomial order on R m , and let P ∈ R k × m be a minimal ≼ -Gröbner basis of Syz N ( F ) . If P is modified by truncating each of itspolynomials modulo ⟨ X d + , . . . , X d r + r ⟩ , then P is still a minimal ≼ -Gröbner basis of Syz N ( F ) . Proof. On the first hand, this modification of P does not af-fect the ≼ -leading terms since they all have X i -degree less than d i + ⟨ lm ≼ ( P )⟩ = ⟨ lm ≼ ( Syz N ( F ))⟩ . On the other hand, after thismodification we also have ⟨ P ⟩ ⊆ Syz N ( F ) since we started from abasis of Syz N ( F ) and added to each of its elements some multiplesof ⟨ X d + , . . . , X d r + r ⟩ , which are contained in Syz N ( F ) . Then [13,Lem. 15.5] yields ⟨ P ⟩ = Syz N ( F ) , hence the conclusion. □ Then, the divide and conquer approach can be refined as de-scribed in Algorithm 4. The correctness of this algorithm can beshown by following the proof of Theorem 4.3 and with the follow-ing considerations. By induction hypothesis, Q is such that eachcomponent of the rows of Q G is an element of ⟨ X d , . . . , X d j − j − , X ⌊ d j / ⌋ j , X j + , . . . , X r ⟩ , hence its truncation modulo ⟨ X d , . . . , X d j j , X j + , . . . , X r ⟩ Algorithm 4
Padé ( d , . . . , d r , G , ≼ , K ) Input: • integers d , . . . , d r ∈ Z > ,• a matrix G in R k × n ,• a monomial order ≼ on R m ,• a list K = ( µ , . . . , µ k ) of elements of Mon (R m ) . Output: • a matrix Q in R ℓ × m for some ℓ ≥ L of ℓ elements of Mon (R m ) . if d = · · · = d r = then Q ∈ R k × k ← I k ; H ← G mod X , . . . , X r ; L ← K for i = , . . . , n do φ ← linear functional R n → K defined by φ ( f ) = f i ( ) ( Q i , L ) ← Syzygy_BaseCase ( φ , H , ≼ , L ) Q ← Q i Q mod X , . . . , X r H ← Q i H mod X , . . . , X r return ( Q , L ) j ← max { i ∈ { , . . . , r } | d i > } ( Q , L ) ← Padé ( d , . . . , d j − , ⌊ d j / ⌋ , , . . . , , G , ≼ , K ) G ← X −⌊ d j / ⌋ j ( Q G mod X d , . . . , X d j j , X j + , . . . , X r ) ( Q , L ) ← Padé ( d , . . . , d j − , ⌈ d j / ⌉ , , . . . , , G , ≼ , L ) Q ← Q Q mod X d + , . . . , X d r + r return ( Q , L ) is an R -multiple of X ⌊ d j / ⌋ j . It follows that on Line 11, G is welldefined. Moreover, for p ∈ R m the next equations are equivalent: pQ G = X d , . . . , X d j − j − , X d j j pG = p X −⌊ d j / ⌋ j Q G = X d , . . . , X d j − j − , X ⌈ d j / ⌉ j This justifies the division by X ⌊ d j / ⌋ j at Line 11 and the fact thatthe second call is done with ⌈ d j / ⌉ instead of d j at Line 12.For the complexity analysis, we use Lemma 5.2 to give a boundon the size of the computed Gröbner bases, which differs from thegeneral bound in Lemma 4.2.Corollary 5.4 (of Lemma 5.2). Let N be as in Eq. (6) , let F ∈R m × n , let ≼ be a monomial order on R m , and let P ∈ R k × m be aminimal ≼ -Gröbner basis of Syz N ( F ) . Then, k ≤ md · · · d r /( max ≤ i ≤ r d i ) . Proof. Let L = lm ≼ ( P ) ∈ R k × m and let ¯ ı be such that d ¯ ı = max ≤ i ≤ r d i . It is enough to prove that L has at most d · · · d r / d ¯ ı rows of the form µ e j for each j ∈ { , . . . , m } ; by Lemma 5.2, themonomial µ ∈ Mon (R) has X i -degree at most d i for 1 ≤ i ≤ r . Now,for each monomial ν = X e · · · X e ¯ ı − ¯ ı − X e ¯ ı + ¯ ı + · · · X e r r with e i ≤ d i forall i (cid:44) ¯ ı , there is at most one row µ e j in L such that µ = νX e ¯ ı forsome e ≥
0: otherwise, one of two such rows would divide theother, which would contradict the minimality of P . The number ofsuch monomials ν is precisely d · · · d r / d ¯ ı . □ Here we have D = nd · · · d r , hence the above bound on the cardi-nality of minimal ≼ -Gröbner bases refines the bound in Lemma 4.2as soon as m ≤ n ( r − )( max ≤ i ≤ r d i ) . SSAC ’20, July 20–23, 2020, Athens, Greece Simone Naldi and Vincent Neiger
Proposition 5.5.
For R = K [ X , Y ] , let N = ⟨ X d , Y e ⟩ × · · · × ⟨ X d , Y e ⟩ ⊂ R n , let F ∈ R m × n with deg X ( F ) < d and deg Y ( F ) < e , and let ≼ be amonomial order on R m . Algorithm 4 computes a minimal ≼ -Gröbnerbasis of Syz N ( F ) using O ˜ (( M ω − + Mn )( M + n ) de ) operations in K ,where M = m min ( d , e ) . Proof. According to Corollary 5.4, the number of rows of thematrices Q computed in Algorithm 4 is at most M = m min ( d , e ) . Itfollows that all matrices Q i , Q , Q , Q in the algorithm have at most M rows and at most M columns, and that the matrices G , H , G , G have at most M rows and exactly n columns. Besides, by Kroneckersubstitution [7, Chap. 1 Sec. 8], multiplying two bivariate matricesof dimensions M × M (resp. M × n ) and bidegree at most ( d , e ) costs O ˜ ( M ω de ) (resp. O ˜ ( M ω ( + n / M ) de ) ) operations in K .Let C( m , n , d , e ) denote the number of field operations used byAlgorithm 4; we have C( m , n , d , e ) ≤ C( M , n , d , e ) . First, for e > C( M , n , d , e ) is bounded by C( M , n , d , ⌊ e / ⌋) + C( M , n , d , ⌈ e / ⌉) + O ˜ ( M ω ( + n / M ) de ) . Indeed, there are two recursive calls withparameters ( d , ⌊ e / ⌋) and ( d , ⌈ e / ⌉) , and two matrix products Q G and Q Q to perform; as noted above, the latter products cost O ˜ ( M ω ( + n / M ) de ) operations in K . The same analysis for d > e = C( M , n , d , ) is bounded by C( M , n , ⌊ d / ⌋ , ) + C( M , n , ⌈ d / ⌉ , ) + O ˜ ( M ω ( + n / M ) d ) .Finally, for d = e =
1, we show that C( M , n , , ) ∈ O ( M ( M + n ) n ) .In this case, there are n iterations of the loop. Each of them makesone call to Syzygy_BaseCase, which uses O ( M ) field operationsfor computing the λ i ’s at Line 7; note that the α j ’s are zero in thepresent context where the linear functional φ corresponds to theconstant coefficient. The computed basis Q i has a single nontrivialcolumn (it has the form in Eq. (3)), so that computing Q i Q mod ⟨ X , . . . , X r ⟩ (resp. Q i H mod ⟨ X , . . . , X r ⟩ ) can be done naively ata cost of O ( M ) (resp. O ( M ( M + n )) ) operations in K .Based on the previous inequalities, unrolling the recursion byfollowing the divide-and-conquer scheme leads to the announcedcomplexity bound. □ ACKNOWLEDGMENTS
Acknowledgements.
The first author acknowledges support fromthe Fondation Mathématique Jacques Hadamard through the Pro-gramme PGMO, project number 2018-0061H.
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