Guessing Gr{ö}bner Bases of Structured Ideals of Relations of Sequences
aa r X i v : . [ c s . S C ] S e p Guessing Gr ¨obner Bases of Structured Ideals of Relations ofSequences
J´er´emy Berthomieu, Mohab Safey El Din
Sorbonne Universit´e,
CNRS , LIP6 , F-75005, Paris, France
Abstract
Assuming su ffi ciently many terms of a n -dimensional table defined over a field are given, weaim at guessing the linear recurrence relations with either constant or polynomial coe ffi cientsthey satisfy. In many applications, the table terms come along with a structure: for instance,they may be zero outside of a cone, they may be built from a Gr¨obner basis of an ideal invariantunder the action of a finite group. Thus, we show how to take advantage of this structure to bothreduce the number of table queries and the number of operations in the base field to recover theideal of relations of the table. In applications like in combinatorics, where all these zero termsmake us guess many fake relations, this allows us to drastically reduce these wrong guesses.These algorithms have been implemented and, experimentally, they let us handle examples thatwe could not manage otherwise.Furthermore, we show which kind of cone and lattice structures are preserved by skew poly-nomial multiplication. This allows us to speed up the guessing of linear recurrence relationswith polynomial coe ffi cients by computing sparse Gr¨obner bases or Gr¨obner bases of an idealinvariant under the action of a finite group in a ring of skew polynomials. Keywords:
Linear recurrence relations, Gr¨obner bases, Symmetries, Change of orderings
1. Introduction
Problem statement and motivations.
Computing or guessing linear recurrence relations satisfiedby a table is a fundamental problem in coding theory for cyclic codes [8, 25] of dimension n ≥
1, combinatorics and computer algebra for solving sparse linear systems, performing sparsepolynomial interpolation, polynomial least-square approximation and Gr¨obner bases changes oforderings in n ≥ ffi cients in the indices allows us to predict the growth of its terms, to classify the di ff erentialnature of their generating series or to evaluate said generating series [29].Depending on the context, an upper bound on the number of table terms might be known inorder to guess these relations. For instance, in coding theory, this is related to the length and the ∗ Laboratoire d’Informatique de Paris 6, Sorbonne Universit´e, boˆıte courrier 169, 4 place Jussieu, F-75252 ParisCedex 05, France.
Email addresses: [email protected] (J´er´emy Berthomieu), [email protected] (Mohab SafeyEl Din)
Preprint submitted to Journal of Symbolic Computation September 14, 2020 inimum distance of the code. In the Gr¨obner bases change of orderings application, an upperbound is given by the degree of the ideal and the number of variables. Whenever no upper boundis known, one is still restricted to only consider a finite number of table terms to guess the linearrecurrence relations the table satisfies. Thus, some of these relations may be proven incorrectwhen tested with many more table terms; further such relations will be called fake relations .This happens for instance in combinatorics where the nature itself of the table may be unknown.In many applications, the table comes with a structure. For instance, in combinatorics,for n D-space walks in the nonnegative orthant, v i , i ,..., i n counts the number of ways to reach( i , . . . , i n ) ∈ N n in i steps of size 1 [9–13]. Therefore, v i , i ,..., i n is trivially 0 outside the cone i , . . . , i n ≤ i . Thus, computationwise, not considering these terms would reduce the size of thetable and thus might be beneficial for guessing the linear recurrence relations satisfied by thetable. Hence, the goal is to exploit this structure to both reduce the number of table queries andthe number of operations to guess the Gr¨obner basis of the ideal of relations. Prior results.
We distinguish two cases: the one-dimensional case, where tables are with oneindex, and the multidimensional one, where tables have n > D terms of a table, the Berlekamp–Massey al-gorithm [3, 28] guesses the linear recurrence relations with constant coe ffi cients of smaller or-der they satisfy while the Beckermann–Labahn algorithm [1] guesses minimal ones with poly-nomial coe ffi cients. Using fast extended Euclidean algorithm, these algorithms can do so in O ( M ( D ) log D ) operations in the base field [1, 14], where M ( D ) = O ( D log D log log D ) [15] is acost function for multiplying two polynomials of degree at most D .In the multidimensional case, several algorithms were designed for guessing linear recurrencerelations with constant coe ffi cients satisfied by the first terms of the tables using linear algebraroutines. For instance, the Berlekamp–Massey–Sakata algorithm [31–33], the S calar -FGLMalgorithm [4, 5] or the A rtinian G orenstein border bases algorithm [30]. Given su ffi cientlymany terms, the first two return a Gr¨obner basis of the ideal of relations while the third one returnsa border basis of this ideal. Furthermore, in [7] the authors designed an algorithm extending boththe Berlekamp–Massey–Sakata and the S calar -FGLM algorithms using polynomial arithmeticand in [6], they extended the S calar -FGLM algorithm for guessing relations with polynomialcoe ffi cients. However, none of all these algorithms were designed to take the structure of thetable terms into account.Gr¨obner bases are the output of several algorithms for guessing linear recurrence relationsand are a fundamental tool in polynomial systems solving. In many applications, polynomialssystems come with a structure, for instance they span an ideal globally invariant under the actionof a finite group G or their supports are in a cone. From the table viewpoint, these are related toonly considering table terms lying either on a lattice [26] or in a cone.In [23], the authors show that for such an ideal, Gr¨obner bases computations through theF [17], F [18] and FGLM [19] algorithms can be sped up with a factor depending on | G | ,whenever the characteristic of the field of coe ffi cients does not divide | G | . To do so, they es-sentially perform | G | parallel smaller computations. In particular for the FGLM algorithm, thisfactor is | G | , see [23, Theorem 10]. Likewise, in [34], the author proposed algorithms for com-puting Gr¨obner bases of symmetric ideals over the rationals or a finite field. In [2, 22], theauthors show that if C is a semi-group of Z n containing 0 and no pair of opposite elementsand if I is an ideal spanned by polynomials with support in the corresponding monomial set T ( C ) ≔ n x i · · · x i n n (cid:12)(cid:12)(cid:12) ( i , . . . , i n ) ∈ C o , then one can modify Gr¨obner bases computation algorithmsto compute generators that behave like a Gr¨obner basis of the ideal but still with all their mono-2ials in T ( C ). This allows them to speed up Gr¨obner basis computations by taking into accountthe sparsity of the union of the supports of the original generators of the ideal. Main results.
We design variants of the S calar -FGLM algorithm guessing linear recurrencerelations for a n -dimensional table v , given as polynomials in x , . . . , x n . The original algorithmis recalled in page 9. In this first theorem, we only consider terms of the table lying on a cone inorder to guess linear recurrence relations with support in the same cone. Theorem 1.1.
Let C be a submonoid cone of N n spanned by the minimal set of generators { a , . . . , a ν } . Let ≺ be a monomial ordering on T , the set of monomials in n variables, and letT ⊂ T ( C ) be finite, stable by division and ordered for ≺ .Then, the S calar -FGLM algorithm called on table v , T and ≺ returns a set of polynomialsG with support in T ( C ) , such that for all s ∈ T \ h lm ≺ ( G ) i , s is in the associated staircase of asparse Gr¨obner basis the ideal of C-relations of v for ≺ .Furthermore, if the ideal of C-relations of v is -dimensional and has a reduced sparseGr¨obner basis with support in T for ≺ , then the output of the S calar -FGLM algorithm called on v and T is this reduced sparse Gr¨obner basis. Let us remark that this allows us to remove trivial constraints on the relations when the tableterms are 0 outside the cone, yielding in practice many fewer guessed relations that eventuallyfail. As a byproduct, this allows us to reduce the number of table queries to guess the relations.For instance, for a subtable of the Gessel walk, using 3 491 table terms, we can guess 136 re-lations amongst which 133 are fake and only 6 are correct. In the meantime, using 3 010 tableterms in a cone, we guess 21 relations and all of them are correct. We refer to Table 2 for moredetails.In the next two theorems, we now consider table terms lying on a lattice Λ and a ffi ne trans-lates thereof. This allows us to design parallel variants of the S calar -FGLM and A daptive S calar -FGLM algorithms. They essentially deal with L multi-Hankel matrices of sizes roughlydivided by L , where L is the number of integer points in the fundamental domain of Λ . We shalldenote T (cid:0) ( a + Λ ) ≥ (cid:1) ≔ n x i · · · x i n n (cid:12)(cid:12)(cid:12) ( i , . . . , i n ) ∈ ( a + Λ ) ∩ N n o , the set of monomials whose non-negative exponents lie in the a ffi ne translate of Λ passing through a . The parallel variant of theS calar -FGLM algorithm, called the L attice S calar -FGLM algorithm, is given in page 12. Theorem 1.2.
Let Λ be a sublattice of Z n with fundamental domain A . Let ≺ be a monomialordering on T and let T ⊂ T be finite, stable by division and ordered for ≺ .Then, the L attice S calar -FGLM algorithm called on table v , T and ≺ returns a truncatedGr¨obner basis of an ideal whose polynomials are each with support in { } ∪ T (cid:0) ( a + Λ ) ≥ (cid:1) .Furthermore, let G be a Gr¨obner basis for ≺ satisfying this support property and with supportin T ⊂ T . Let S be the associated staircase and v be a generic C-finite table whose ideal ofrelations is spanned by G . Then, there exists a non empty Zariski open set of values for the tableterms [ s ] of v , with s ∈ S = { tt ′ , t , t ′ ∈ S} , such that the L attice S calar -FGLM algorithm calledon v , ≺ , T and A correctly guesses G . For a monomial m = x i · · · x i n n and a n -dimensional table v , let us denote [ m ] = h x i · · · x i n n i = u i ,..., i n . We extend this notation linearly to polynomials: for a polynomial g = P ( i ,..., i n ) g i ,..., i n x i · · · x i n n and table v , we let (cid:2) g (cid:3) = P ( i ,..., i n ) g i ,..., i n h x i · · · x i n n i = P ( i ,..., i n ) g i ,..., i n u i ,..., i n . This is the evaluation of g as a linear combination of terms of v . 3he parallel variant of the A daptive S calar -FGLM algorithm, called the L attice A daptive S calar -FGLM algorithm, is given in page 18. Theorem 1.3.
Let Λ be a sublattice of Z n with fundamental domain A . Let ≺ be a monomialordering on T . Let us assume that the L attice A daptive S calar -FGLM algorithm called ontable v , ≺ , Λ and A ⊆ N n returns a set of polynomials G.Let us denote by S the associated staircase to G and S a = S ∩ T (cid:0) ( a + Λ ) ≥ (cid:1) for each a ∈ A .Then, for any polynomial g ∈ G with lm ≺ ( g ) ∈ T (cid:0) ( a + Λ ) ≥ (cid:1) and any s ∈ S a with s ≺ lm ≺ ( g ) ,we have (cid:2) gs (cid:3) = .Furthermore, let G be a Gr¨obner basis for ≺ such that for all g ∈ G, there exists a ∈ A such that supp g ⊂ T (cid:0) ( a + Λ ) ≥ (cid:1) . Let S be the associated staircase and v be a generic C-finitetable whose ideal of relations is spanned by G . Then, there exists a non empty Zariski open setof values for the table terms [ s ] of v , with s ∈ S = { tt ′ , t , t ′ ∈ S} , such that the L attice A daptive S calar -FGLM algorithm called on v , ≺ and A correctly guesses G .Structure of the paper. We first recall in Section 2 the classical connection between linear re-currence relations with polynomial coe ffi cients and skew polynomials in 2 n variables. Then,we recall how using linear algebra routines on a special kind of matrix, a multi-Hankel one,the S calar -FGLM algorithm, and its adaptive variant the A daptive S calar -FGLM algorithm,guesses linear recurrence relations.In Section 3, we design variants of the S calar -FGLM algorithm that take the table structureinto account for guessing linear recurrence relations, then we prove Theorems 1.1 and 1.2. Asan application, we provide a modification of the S parse -FGLM algorithm [20, 21] whenever theideal is globally invariant under the action of a finite group.The same kind of variants of the A daptive S calar -FGLM algorithm are then designed, inSection 4. Likewise, we prove Theorem 1.3 in this section. Then, we show how one can performskew polynomial operations in order to preserve the cone and lattice structures of the support ofthe polynomials.Finally, in Section 5, we report on our speedup using our C implementation of the S parse -FGLM algorithm when the ideal is invariant under the action of a finite group. We also guesslinear recurrence relations satisfied by n D-space walks with and without exploiting the conestructure of the table and then test further the guessed relations. We then report on how the conestructure allows us to guess fewer fake linear recurrence relations.
2. Preliminaries
Let N be the set of natural numbers, and Z be the ring of integers. For n ∈ N , n ≥
1, welet i = ( i , . . . , i n ) ∈ N n , x = ( x , . . . , x n ) and x i = x i · · · x i n n . For a subset S of N n , we let T ( S ) = { x s | s ∈ S} be the set of monomials with exponents in S . To ease the presentation, welet T ≔ T ( N n ). Finally, for a polynomial f = P s ∈S f s x s , we let supp f = { s ∈ S| f s , } be itssupport.Let K be a field and v ∈ K N n be a n -indexed sequence with values in K , that is v = ( v i ,..., i n ) ( i ,..., i n ) ∈ N n . There is a natural correspondence between finite linear combinations of termsof v and polynomials in K [ x , . . . , x n ]. For g = P s ∈S γ s x s , with S a finite subset of N n , we can4rite (cid:2) g (cid:3) ≔ P s ∈S γ s v s . Hence shifting a relation by an index i comes down to multiplying thecorresponding polynomial by x i since h g x i i = X s ∈S γ s v s + i . In particular, a polynomial g defines a linear recurrence relation with constant coe ffi cients , or C-relation for short, on v if, and only if, for all i ∈ N n , h g x i i =
0. The set of all such polynomialsis an ideal of K [ x ] called the ideal of C-relations of v , see for instance [5, Definition 2 andProposition 4].Finally, a nonzero sequence v is said to be C-finite if together with a finite number of termsof v and a finite number of C-relations, one can recover all the terms of v . This is equivalent torequiring that the ideal of C-relations of v is 0-dimensional, see also [24, Definition 2.2]. Example 2.1.
On the one hand, the terms v i , j = (5 + i + j )2 i + j + (3 + i + j )5 i + j of v ∈ F N canall be computed thanks to v , = v , = v , = , v , = and the C-relations, for all ( i , j ) ∈ N ,v i + , j + + v i , j = v i + , j + v i , j + + v i , j = v i , j + + v i + , j + v i , j + = . On the other hand, they can also be computed knowing v , = v , = v , = v , = and that forall ( i , j ) ∈ N , v i , j + + v i , j + + v i , j = v i + , j + v i , j + + v i , j + = . Thus, the ideal of C-relations of v is the -dimensional one D xy + , x + y + , y + x + y E = D y + y + , x + y + y E and v is C-finite.On the other hand, the binomial sequence, b = (cid:16) b i , j (cid:17) ( i , j ) ∈ N = (cid:16)(cid:16) ij (cid:17)(cid:17) ( i , j ) ∈ N , satisfies Pascal’srule: for all ( i , j ) ∈ N , b i + , j + − b i , j + − b i , j = . Moreover, one can show that this relationspans all the other C-relations, i.e. its ideal of C-relations is the -dimensional one h xy − y − i ,thus b is not C-finite. Furthermore, some sequences satisfy linear recurrence relations with coe ffi cients that arepolynomials in the indices of the sequence, or P-relations for short. For instance, the binomialsequence satisfies the following two P-relations for all ( i , j ) ∈ N :( j + b i , j + − ( i − j ) b i , j = i + − j ) b i + , j − ( i + b i , j = . Combining them by shifting the former by index (0 ,
1) and then adding the latter yields( i − j ) b i + , j + − ( i − j ) b i , j + − ( i − j ) b i , j = . This proves that Pascal’s rule holds whenever i , j .We thus aim at representing the former relations as polynomials g and g such that for all( i , j ) ∈ N , h g x i y j i = h g x i y j i =
0. For instance, we could say that the first one correspondsto h ( j + x i y j + − ( i − j ) x i y j i = h (( j + y − ( i − j )) x i y j i =
0, but this would mean that g hascoe ffi cients in i and j , which are meaningless on their own. To circumvent this, in [6], the authorsintroduced a new set of variables t = ( t , . . . , t n ), such that t p behaves like x p ∂ p , where ∂ p is thedi ff erential operator with respect to x p . That is, h t k x i i ≔ h ( x ∂ , . . . , x n ∂ n ) k x i i = h i j · · · i j n n x i i = j · · · i j n n v i = i k v i . Then, the [ . ] notation is naturally K -linearly extended to polynomials in t and x . Therefore, the 2 n variables t , . . . , t n , x , . . . , x n follow, for all 1 ≤ p , q ≤ n and p , q ,the commutation rules x p x q = x q x p , t p t q = t q t p , t p x q = x q t p and t p x p = x p ( t p + t and x quasi-commutative . The ring of skew polynomials in t and x will bedenoted K h t , x i while the ring of skew polynomials in x with coe ffi cients in K ( t ) will simply bedenoted K ( t ) h x i . Now, a P-relation is given by a finite subset S of N n and polynomials γ s ∈ K [ i ]for s ∈ S , such that ∀ i ∈ N n , X s ∈S γ s ( s + i ) v s + i = . This relation corresponds to the polynomial g = P s ∈S γ s ( t ) x i ∈ K h t , x i such that for all i ∈ N n , h g x i i = Remark 2.2.
While we can obviously write P s ∈S ˜ γ k ( i ) v s + i = , the former notation with γ s ( s + i ) makes more explicit the relationship with the corresponding polynomial in K h t , x i . Example 2.3.
Let t = t , u = t , x = x and y = x . Then, the P-relations satisfied by thebinomial sequence can be rewritten as ( j + b i , j + − ( i − j ) b i , j = h ( j + x i y j + − ( i − j ) x i y j i = h ux i y j + − ( t − u ) x i y j i = h ( uy − ( t − u )) x i y j i and ( i + − j ) b i + , j − ( i + b i , j = h ( i + − j ) x i + y j − ( i + x i y j i = h ( t − u ) x i + y j − ( t + x i y j i = h (( t − u ) x − ( t + x i y j i . Thus, g = uy − ( t − u ) and g = ( t − u ) x − ( t + in K h t , u , x , y i . The set of all such polynomials is a right ideal of K h t , x i . Indeed, it is stable by multiplicationon the right by any monomial x i as requested. Furthermore, since t ℓ x j t k x i = t ℓ ( t − j ) k x j + i , then h t ℓ x j t k x i i = h t ℓ ( t − j ) k x j + i i = ( j + i ) ℓ i k v j + i = i k h t ℓ x j + i i . In other words, multiplying on theright by t k x i corresponds to multiplying on the right by x i and to multiply the evaluation by aconstant, namely i k . Thus if h g x i i vanishes, then so does h g t k x i i .Analogously to the C-finite definition, a nonzero sequence v is said P-finite if a finite numberof its terms and a finite number of P-relations allows one to recover all of its terms, though,one has to deal with singularities coming from integers roots of the leading coe ffi cients of therelations. This is equivalent to requiring that the sequence v and the ideal of P-relations of v is0-dimensional in K ( t ) h x i while also dealing with the singularities given by the denominators ofthe coe ffi cients. Example 2.4 (Cont. of Example 2.3) . The ideal of P-relations of b in K h t , u , x , y i is h uy − ( t − u ) , ( t − u ) x − ( t + , xy − y − i . Furthermore, since ( xy − y −
1) ( t − u ) = ( t − u ) xy − ( t + − u ) y − ( t − u ) = (( t − u ) x − ( t + y + ( uy − ( t − u )) , then, in K ( t , u ) h x , y i , its ideal of P-relations is only spanned by uy − ( t − u ) and ( t − u ) x − ( t + . .2. Gr¨obner bases This section briefly recalls some basic definitions on Gr¨obner bases. The interested readerwill find more details in [16].For T the set of monomials in K h t , x i , a monomial ordering ≺ on T is an order relationsatisfying the following three properties1. ∀ m ∈ T , 1 (cid:22) m ;2. ∀ m , m ′ , s ∈ T , m (cid:22) m ′ ⇒ ms (cid:22) m ′ s .For a monomial ordering ≺ on K h t , x i , the leading monomial of f , denoted lm ≺ ( f ), or lm ≺ ( f )if there is no ambiguity on ≺ , is the greatest monomial in the support of f for ≺ . For an ideal I ,we let lm ≺ ( I ) = { lm ≺ ( f ) , f ∈ I } . We recall briefly the definition of a Gr¨obner basis and of itsassociated staircase. Definition 2.5.
Let I be a nonzero ideal of K h t , x i and let ≺ be a monomial ordering. A set G ⊆
I is a
Gr¨obner basis of I if for all f ∈ I, there exists g ∈ G such that lm ≺ ( g ) | lm ≺ ( f ) , it is reduced if for any g , g ′ ∈ G , g , g ′ and any monomial m ∈ supp g ′ , lm ≺ ( g ) ∤ m.The staircase of G is defined as S = Staircase( G ) = { s ∈ T , ∀ g ∈ G , lm ≺ ( g ) ∤ s } . It is also thecanonical basis of K h t , x i / I as a K -vector space. Gr¨obner basis theory allows us to choose any monomial ordering, among which we mainlyuse, on the x variables, the lex ( x n ≺ · · · ≺ x ) ordering which satisfies x i ≺ x j if, and only if, there exists 1 ≤ p ≤ n suchthat for all q < p , i q = j q and i p < j p , see [16, Chapter 2, Definition 3]; drl ( x n ≺ · · · ≺ x ) ordering which satisfies x i ≺ x j if, and only if, i + · · · + i n < j + · · · + j n or i + · · · + i n = j + · · · + j n and there exists 2 ≤ p ≤ n such that for all q > p , i q = j q and i p > j p , see [16, Chapter 2, Definition 6].We will also use monomial orderings on the t and x variables. Since we want to freely switchfrom K h t , x i to K ( t ) h x i and vice versa, it makes sense to choose an ordering such that t k ≺ x ℓ for any k and ℓ , such as lex ( t n ≺ · · · ≺ t ≺ x n ≺ · · · x ) or drl ( t n ≺ · · · ≺ t ≺ · · · ≺ x ). Thelatter is more suitable as it allows us to enumerate all the monomials in t and x in increasingorder. A cone C is a subset of Z n such that if i ∈ C , then for every λ ∈ N , λ i ∈ C . The cones weare interested in are those that are submonoids of N n , i.e. 0 ∈ C and for all i , j ∈ C , ( i + j ) ∈ C ,containing no opposite elements but 0, that is if i ∈ C \ { } , then − i < C .Given such a cone C and polynomials with support in its associated set of monomials T ( C ) = n x i ∈ T (cid:12)(cid:12)(cid:12) i ∈ C o , one may want to perform all the polynomial operations in T ( C ) in order to takeadvantage of the structure of the support when computing a Gr¨obner basis of the ideal they span.This leads to the definition of sparse Gr¨obner basis with support in T ( C ) that uses its monoidstructure. Definition 2.6 ([22, Definition 3.1] and [2, Definition 3.3]) . Let
C ⊆ N n be a cone and T ( C ) beits associated set of monomials. Let I = h f , . . . , f s i ⊆ K [ x ] be a polynomial ideal such that forall k, the support of f k is in T ( C ) . Then, a sparse Gr¨obner basis of I for a monomial ordering ≺ isa generating set G = { g , . . . , g r } such that for all k, the support of g k is in T ( C ) and for all f ∈ Iwith support in T ( C ) , lm ≺ ( f ) = lm ≺ ( g ) m for some g ∈ G and m ∈ T ( C ) . C = N n , sparse Gr¨obner bases are classical Gr¨obner bases. Furthermore,a sparse Gr¨obner basis of an ideal I for C allows one to determine if a polynomial f , with supportin T ( C ), is in I .For a lattice Λ ⊆ Z n , we let Λ ≥ = Λ ∩ N n be its nonnegative cone, so that, naturally, Z n ≥ = N n . In particular, Λ and Λ ≥ are cones and we may intersect them with another cone. For a ∈ Z n , we also denote by a + Λ the a ffi ne lattice obtained by translating Λ by a and likewiseconsider its intersection with a cone. In particular, ( a + Λ ) ≥ = ( a + Λ ) ∩ N n .Given a lattice Λ , its a ffi ne translates a + Λ = Λ , . . . , a L + Λ and polynomials f , . . . , f k , eachwith supports in an associated set of monomials T (( a ℓ + Λ ) ≥ ), then a reduced Gr¨obner basis of h f , . . . , f k i satisfies also this support property. This allows one to speed up the Gr¨obner basescomputations by essentially performing L computations in parallel with input of sizes divided by L . Given a table v and a polynomial g ∈ K h t , x i , in order to check if g is is in the ideal ofP-relations of v , one must check that h g x i i = i . As only a finite number of terms of v are known, only a finite number of such tests can be done. Now, if only a superset of thesupport of g is known, one can find all the candidates for g by solving a linear system where eachcolumn corresponds to a monomial t ℓ x j , each row a monomial x i and the coe ffi cient at theirintersection is h t ℓ x j + i i = ( j + i ) ℓ v j + i . Such a matrix is called multi-Hankel , a generalization ofHankel matrices. Indeed a Hankel matrix ( h i , j ) ≤ i , j ≤ d satisfies h i , j = v i + j for some table v . Example 2.7.
Let v = ( v i , j ) ( i , j ) ∈ N be a table and T = { , u , t , y , x , uy , ty , ux , tx } ⊂ T (cid:16) N n (cid:17) andX = n , y , x , y , xy , x o ⊂ T ( N n ) be two sets of monomials, then their multi-Hankel matrix isH X , T = u t y x uy ty ux tx v , v , v , v , v , y v , v , v , v , v , v , v , x v , v , v , v , v , v , v , y v , v , v , v , v , v , v , xy v , v , v , v , v , v , v , v , v , x v , v , v , v , v , v , v , . We give some computation details. The coe ffi cient on the third column (t) and first row ( ) is [ t × = h tx y i = v , = . Likewise, the coe ffi cient on sixth column (uy) and the second tolast row (xy) is (cid:2) uyxy (cid:3) = h uxy i = v , = v , .Note that rows are only indexed with monomials in x and not in t , x since the row labeledwith t k x i , k , would be a multiple of the row labeled with x i .2.5. The S calar -FGLM algorithm The S calar -FGLM algorithm [4, 5], takes as an input the table v and a set of monomials T stable by division and computes the kernel of the multi-Hankel matrix H T , T . Vectors in thiskernel can be seen as polynomials in K [ x ]. Those with a leading term minimal for the partialorder induced by the division form the target Gr¨obner basis. If T is ordered for a monomialordering ≺ and contains the staircase and the leading monomials of the reduced Gr¨obner basis ofthe ideal of relations of v for ≺ , then the S calar -FGLM algorithm returns this Gr¨obner basis.8 lgorithm 1: S calar -FGLM Input:
A table v = ( v i ) i ∈ N n with coe ffi cients in K , a monomial ordering ≺ , a su ffi ciently large set ofmonomials T stable by division and ordered for ≺ . Output:
A reduced Gr¨obner basis of the ideal of C-relations of v .Build the matrix H T , T .Compute the set S ⊆ T of smallest monomials, for ≺ , such that rank H S , S = rank H T , T . For all m ∈ T \ S do // stabilize S for the division If ∃ s ∈ S such that m | s then S ≔ S ∪ { m } . L ≔ T \ S sorted for ≺ . G ≔ ∅ . While L , ∅ do g ≔ min ≺ L Solve the linear system H S , S γ + H S , { g } = G ≔ G ∪ (cid:8) g + P s ∈ S γ s s (cid:9) .Remove g and any of its multiples from L . Return G . As our goal is to extend the S calar -FGLM algorithm in order to deal with table terms lyingon a cone or a lattice, we recall this algorithm below.The algorithm computes the column rank profile of the matrix H T , T , that is the set of left-most linearly independent columns of the matrix. Since these columns are independent from theprevious ones, then their labels cannot be the leading monomial, for ≺ , of any polynomial in theideal of C-relations, thus they are in the associated staircase of the reduced Gr¨obner basis of thisideal for ≺ . If T is not large enough, a monomial m could be detected as not lying in the staircasewhile one of its multiples does, hence there is a stabilization process to add m to the staircase ifthis happens. Then, each polynomial in the output Gr¨obner basis is computed by solving a linearsystem involving its leading monomial and the monomials in the staircase. Example 2.8 (Cont. of Example 2.1) . Let us recall that a Gr¨obner basis of the ideal of C-relation of v is n xy + , x + y + , y + x + y o for drl ( y ≺ x ) , hence this ideal has degree .Therefore, the staircase of the Gr¨obner basis of this ideal for lex ( y ≺ x ) , or any monomialordering, can only contain monomials x i y j with ( i + × ( j + ≤ and it su ffi ces to takeT = n , y , y , y , y , x , xy , x , x , x o to recover the staircase and the Gr¨obner basis. The columnrank profile of H T , T is given by S = n , y , y , y o so that L = n y , x , xy , x , x , x o . Then, the linearsystems H S , S γ + H S , { y } = and H S , S γ + H S , { x } = yield the Gr¨obner basis n y + y + , x + y + y o . In many applications, for instance the S parse -FGLM algorithm one, the computation of asingle table element is costly. Therefore, we may want to reduce the number of table queriesperformed by the S calar -FGLM algorithm. In the original algorithm, described in Algorithm 1,it requires T table terms, where 2 T is the Minkowski Minkowski sum of T with itself. To doso, the goal is to let the multi-Hankel grow step by step. It starts with the 1 × (cid:16) [1] (cid:17) .
9f [1] = v ,
0, then 1 is in the associated staircase of the Gr¨obner basis of the ideal of C-relations of v , otherwise it stops and return the Gr¨obner basis { } . The algorithm extends afull-rank matrix H S , S into H S ∪{ m } , S ∪{ m } with m greater, for ≺ , than any monomial in S . Now,there are two possibilities, either the new matrix is full rank or it is not and the column labeledwith m is linearly dependent from the other ones. In the former case, m is actually in this staircaseand S is replaced by S ∪ { m } . In the latter case, a polynomial with support in S ∪ { m } and leadingmonomial, for ≺ , m is found and no multiples of m will ever be proposed to extend the multi-Hankel matrix. The algorithm stops either when no monomials can be added to the staircase orwhen the size of staircase has reached a threshold given in input. There is, however, a possibilityof finding wrong relations if the first terms of the table exceptionally satisfies a relation of smallerorder, for instance if v =
0. This problem can be circumvented by testing relations further if therelations are suspiciously too small, for instance in FGLM applications where the degree of theideal is known in advance.
3. Guessing with structures
In this section, we show how to guess linear recurrence relations of a table by taking thestructure of the table terms into account. We first start with the case where only table terms ina cone are considered. Then, we study how to guess these relations when table terms are in alattice or some a ffi ne translates thereof. In this subsection, we aim at describing how we can take advantage of the structure of a givencone C to recover the ideal of relations of a table v by only considering table terms inside thecone. That is, we aim at guessing polynomials g such that supp g ⊂ T ( C ) and for all x i ∈ T ( C ), h g x i i =
0. The latter condition is the guessing part as we will only be able to ensure that h g x i i = x i in a finite subset T of T ( C ).To do so, two strategies are at our disposal and they both rely on the generators of C as asubmonoid of N n . Let us denote by a , . . . , a ν a set of generators of C , i.e. for all i ∈ C , thereexists j ∈ N ν such that i = j a + · · · + j ν a ν . First and foremost, there is no reason for ν ≤ n andsecond, even if ν is minimal and a , . . . , a ν is a generating set, there is no reason for ( j , . . . , j ν )to be unique. Example 3.1.
The cone C = n i ∈ N (cid:12)(cid:12)(cid:12) i ≤ i , i ≤ i o is spanned by a = (1 , , a = (1 , and a = (2 , so that C = n j a + j a + j a (cid:12)(cid:12)(cid:12) a = (1 , , a = (1 , , a = (2 , o . Yet, we have thetwo decompositions (3 , = a = a + a . The first strategy is designed to only consider table terms lying in C . Assuming a generatingset a , . . . , a ν of C is known, the set of monomials T ( C ) can be defined as T ( C ) = n x j a · · · x j ν a ν (cid:12)(cid:12)(cid:12) ( j , . . . , j ν ) ∈ N ν o . The second strategy make use of a new set of variables y = ( y , . . . , y ν ), so that y represents x a ,etc and an auxiliary table w = ( w j ) j ∈ N ν defined by w j = v j a + ··· + j ν a ν . Then, two monomials y j and y k represent the same monomial x i if, and only if, i = j a + · · · + j ν a ν = k a + · · · + k ν a ν . Thatis, both w j and w k are equal to w i . Thus, w satisfies extra relations coming from these multiple10quivalent writings. They are given by binomials, namely y j − y k . Hence, not all monomials in T ( N ν ) are of interest and we clean them up by using the binomial ideal I ( C ) they span.In practice, both strategies are equivalent. They only di ff er in how they enumerate table terms v i with i ∈ C . Even though, this should not be the bottleneck, either the linear algebra routinesor the computations of the terms should be, the second strategy requires computing a Gr¨obnerbasis of I ( C ), for instance using [27] while the first one only requires checking that a monomialhas already been generated.Since the first strategy comes down to directly calling the S calar -FGLM algorithm with aset of monomials T ⊂ T ( C ), this yields Theorem 1.1. Proof of Theorem 1.1.
As the S calar -FGLM algorithm computes kernel vectors of H T , T , thecorresponding polynomials can only have support in T ( C ).Let S be the associated staircase of a sparse Gr¨obner basis of the ideal of C-relations of v .Let us show first that no monomial m < S is found in the staircase by the algorithm. As m ∈ lm ≺ ( I ), there exist α s ∈ K , for all s ∈ S such that m + P s ∈ S α s s ∈ I , thus (cid:2) t (cid:0) m + P s ∈ S α s s (cid:1)(cid:3) = t ∈ T . Since T ⊂ T , then this means that column labeled with m is linearly dependentfrom the previous ones and neither m nor any multiples thereof is in the staircase associated tothe output. Hence, the computed staircase is included in the correct staircase.Let us now assume that the ideal of C-relations of v is 0-dimensional, that is S is finite. Weshall show by contradiction that matrix H S , S is full rank, so that the output of the S calar -FGLMalgorithm called on T ⊃ S is a reduced Gr¨obner basis whose associated staircase contains S .Let us assume that H S , S is not full rank and let m < S be the smallest monomial for ≺ suchthat rank H S , S ∪{ m } > rank H S , S . Let R be any finite subset of S ∪ { µ | µ (cid:22) m } stable by divisionand containing S ∪ { m } . By minimality of m , for ≺ , rank H S , R = rank H S , S ∪{ m } > rank H S , S and in particular column labeled with m must be independent from the previous ones. Thus, nopolynomial with leading monomial m can be in the ideal of relations and m is in the staircase ofthis ideal. This is a contradiction with the assumption that m is not in S . Since S ⊆ T , then thealgorithm correctly computes a superset of the staircase S and thus the algorithm discovers thecorrect staircase.Finally, the polynomials of the sparse Gr¨obner basis are found by linear algebra.The second strategy comes down to calling the S calar -FGLM algorithm on w with a set ofmonomials T ⊆ T ( N ν ) / I ( C ). Furthermore, it is clear that T ( N ν ) / I ( C ) is stable by division, thuswe can always call the S calar -FGLM algorithm with T stable by division. Then, by construction,it remains to replace the polynomials obtained in K [ y ] by the corresponding ones in K [ x ]. Theywill naturally have support in T ( C ). Example 3.2 (Continuation of Example 3.1) . It is clear that a = a + a generates all theother di ff erent ways to decompose an element of C , hence I ( C ) = D y − y y E . Thus, when listingthe monomials for drl ( y ≺ y ≺ y ) in T ( N ν ) / I ( C ) , we will skip any multiple of y .3.2. Terms in a lattice Let Λ ≥ be the set of nonnegative terms of a sublattice of Z n , we aim at guessing the recur-rence relations of a table v by following Λ ≥ . Since a lattice is a special case of a cone, if werestrict ourselves to only considering the subtable ( v i ) i ∈ Λ ≥ and if we can ensure that the supportof the reduced Gr¨obner basis of the ideal of relations is in T ( Λ ≥ ), then Theorem 1.1 asserts that11e can guess the ideal of relations using the S calar -FGLM algorithm called on T ⊂ T ( Λ ≥ )su ffi ciently large.Yet, doing so would in some way make us forget the extra structure coming with a sublattice:namely its fundamental domain, i.e. the quotient group Z n / Λ . Indeed, if a set of polynomials { f , . . . , f r } satisfies for all k , there exists a k ∈ Z n / Λ such that supp f k ∈ ( a k + Λ ) ≥ , then areduced Gr¨obner basis G = { g , . . . , g s } of the ideal it spans satisfies the same property. There-fore, if we expect, or even can ensure beforehand, that the reduced Gr¨obner basis of the ideal ofrelations of v also satisfies this property, we aim at guessing this Gr¨obner basis by working inparallel on several smaller multi-Hankel matrices whose sizes have been divided by Z n / Λ ).To do so, considering an input set of monomials T ⊂ T , we shall split it up into T = F a ∈ Z n / Λ T a ,with T a = T ∩ T (cid:0) ( a + Λ ) ≥ (cid:1) , and then call the S calar -FGLM algorithm on v and T a for each a .However, the table terms that appear in H T a , T a are v i with i ∈ (2 a + Λ ) ≥ . Thus, we might neverconsider certain table terms. To circumvent this, we always add the row and the column labeledwith 1 in these matrices. This yields the L attice S calar -FGLM algorithm or Algorithm 2 andTheorem 1.2. Algorithm 2: L attice S calar -FGLM Input:
A table v = ( v i ) i ∈ N n with coe ffi cients in K , a monomial ordering ≺ , a set of monomials T stable by division and ordered for ≺ , a nonnegative lattice Λ ⊆ Z n , a set A ⊆ N n containing 0such that Λ + A = Z n . Output:
A truncated reduced Gr¨obner basis.Partition T into T = F a ∈A T a with T a = (cid:0) T ∩ T (cid:0) ( a + Λ ) ≥ (cid:1)(cid:1) . For all a ∈ A do Build the matrix H { }∪ T a , { }∪ T a .Compute its column profile rank S a . S ≔ S a ∈A S a . For all m ∈ T \ S do // stabilize S for the division If ∃ s ∈ S such that m | s ∈ S then S ≔ S ∪ { m } . L ≔ T \ S sorted for ≺ . G ≔ ∅ . While L , ∅ do g ≔ min ≺ L Find a ∈ A such that g ∈ T (cid:0) ( a + Λ ) ≥ (cid:1) .Solve the linear system H S a , S a γ + H S a , { g } = G ≔ G ∪ n g + P s ∈ S a γ s s o .Remove g and any of its multiples from L . Return G . Proof of Theorem 1.2.
This proof follows mostly the same steps as this of Theorem 1.1.As the algorithm computes kernel vectors of matrices H { }∪ T a , { }∪ T a , the corresponding poly-nomials can only have support in { } ∪ T (cid:0) ( a + Λ ) ≥ (cid:1) .Let S be the associated staircase of a reduced Gr¨obner basis of the ideal of C-relations of v .For each a ∈ A , we let S a = S ∩ T (cid:0) ( a + Λ ) ≥ (cid:1) .Let us show first that no monomial m < S is found in the staircase by the algorithm. As m ∈ lm ≺ ( I ), there exist g = lm ≺ ( g ) + P α s ∈ S a α s s ∈ I such that lm ≺ ( g ) ∈ T (cid:0) ( a + Λ ) ≥ (cid:1) and12 m ≺ ( g ) | m . Thus, m lm ≺ ( g ) g ∈ I and for all t ∈ T , h t m lm ≺ ( g ) g i =
0. In particular, this is true for all t ∈ T b , with m ∈ T b , so that column labeled with m is linearly dependent from the previous onesin H { }∪ T b , { }∪ T b . Hence, neither m nor any of its multiples is in the staircase associated to theoutput. That is, the computed staircase is included in the correct staircase.It remains to prove the last statement. For any a ∈ A , let S a = S∩T (cid:0) ( a + Λ ) ≥ (cid:1) . A necessaryand su ffi cient condition for the L attice S calar -FGLM algorithm to correctly guess G is that foreach a , S a ⊆ S a , which means that either S a = S a or S a = { }∪S a . In particular, if S a = { }∪S a and each submatrix of H S a , S a is full rank, then these conditions are satisfied. Requiring that eachsubmatrix of each matrix H S a , S a is full rank can be turned into a nonzero polynomial system,made of their determinants, whose set of indeterminates is the matrix coe ffi cients, i.e. the tableterms [ s ] with s ∈ S a ∈A S a ⊆ S . Hence, there is a non empty Zariski open set of values forthese table terms such that S and G are correctly guessed. Remark 3.3.
Adding a row labeled with in the matrices is necessary to prevent computationsof incorrect relations when one of them is divisible by a non trivial monomial. Let us considera unidimensional table satisfying the relation x + ax with a ∈ K and let Λ = Z and T = n , x , x , x , x o , so that T = n , x , x o and T = n x , x o . We thus build the matricesH T , T = x x [1] [ x ] [ x ] x [ x ] [ x ] [ x ] x [ x ] [ x ] [ x ] , H T , T = x x x [ x ] [ x ] x [ x ] [ x ] ! . By hypothesis, clearly column labeled with x is linearly dependent from the ones with label and x . However, since h x + ax i = h x + ax i = , then column labeled with x is linearlydependent from the column labeled with x in the second matrix. Therefore, these matrices doesnot allow us to recover that x is in the staircase of the ideal of relations of the input table. Example 3.4.
Consider the table v = (cid:16) i ( j + (cid:17) ( i , j ) ∈ N defined over Q . Using, for in-stance, the Berlekamp–Massey–Sakata or the S calar -FGLM algorithms, we can easily showthat its ideal of relations is D y − , x − E . Let us consider the lattice Λ = (0 , Z + (1 , Z , sothat A = { (0 , , (0 , , (0 , } and T = n , y , y , y , y , y , x o .Then, Algorithm 2 builds the matrices y x y x , y y y y , y y y y . So that S = { } , S = { , y } and S = n , y o . Hence S = n , y , y o , L = n y , y , y , x o . This yieldsthe linear systems H S , S γ + H S , { y } = and H S , S γ + H S , { x } = allowing us to recover y − and x − .Notice that w = (cid:16) i ( j mod 3) (cid:17) ( i , j ) ∈ N has the same ideal of relations. Yet, the algorithm willbuild the matrices y x y x , y y y y , y y y y , o that S = ∅ , S = { , y } , S = n , y o and S = n , y , y o . Since the linear systems H S , S γ + H S , { x } = and H S , S γ + H S , { y } = are empty, they do not allow us to recover x − and y − .Indeed, ∅ = S , S ∩ T ( Λ ≥ ) = { } . Remark 3.5.
While we assume that Λ is a sublattice of Z n , hence of rank n, it can actually be any Z -submodule of smaller rank ν . However, this means we can only guess an ideal of relations in ν variables so that it may not be the whole ideal of relations. Nevertheless, this kind of restrictioncan be of interest in the P-finite application where the kernel equation makes us study the P-finitenature of a subsequence where some indices are set.3.3. Application to the action of a matrix group When applying a Gr¨obner basis change of orderings algorithm, such as [21], the idea is tobuild the multiplication matrix M n of x n for the already known Gr¨obner basis (typically drl ( x n ≺· · · ≺ x )), to pick a random vector r and then to compute the table (cid:16) r T M in (cid:17) ≤ i ≤ D − , where D is the degree of the ideal and the first vector of the canonical basis and to recover the minimalpolynomial of M n (and x n ) using Wiedemann and Berlekamp–Massey algorithms. It works wellbecause M n is actually sparse when the ideal is spanned by generic polynomials.The goal of this section is to extend this approach to group actions on the ideal. In particular,we will restrict ourselves to finite matrix group actions, that is finite subgroups of GL( n ) where A ∈ GL( n ) acts on f ( x ) ∈ K [ x , . . . , x n ] by sending it to f ( A x ). We start by recalling some results on finite matrix group actions on ideals of K [ x ].Since the group G is finite, by the invariant factors theorem, there exist q | · · · | q ℓ such that G ≃ Z / q Z × · · · × Z / q ℓ Z and in particular, for any g ∈ G , G q ℓ = q ℓ is minimal for thisproperty.Furthermore if | G | = q · · · q ℓ is not divisible by the characteristic of the coe ffi cient field K ,then there exists a primitive q ℓ th root of unity ζ such that the matrices in G are simultaneouslydiagonalizable with powers of ζ on the diagonals, see [23, Theorem 2]. After this diagonalizationprocess, which comes down to a linear change of variables, for each matrix in G , there existnatural numbers 0 ≤ ε , . . . , ε n ≤ q ℓ − x i is sent onto ζ ε i x i by this matrix. Definition 3.6 ([23, Definition 3]) . Let G ≃ Z / q Z × · · · × Z / q ℓ Z , with q | · · · | q ℓ , be a diagonalsubgroup of GL( n ) and ζ be a q ℓ th root of unity, then there exist matrices D , . . . , D n spanning Gsuch that each D i has order q i .For each monomial m ∈ T , there exist ( µ , . . . , µ ℓ ) ∈ Z / q Z × · · · × Z / q ℓ Z such that m is sentonto ζ µ i q ℓ / q i m by D i . Then, m is said to have G -degree ( µ , . . . , µ ℓ ) .Furthermore, a polynomial is G -homogeneous if all its monomials have same G-degree. From this, one can prove that the G -degree of the product of two monomials is the sum oftheir G -degrees. Since the G -degree of the monomial 1 is (0 , . . . , G -degree (0 , . . . ,
0) is a sublattice T ( Λ ≥ ) of T . A consequence of this is that if f , . . . , f s are G -homogeneous polynomials, then a reduced Gr¨obner basis of h f , . . . , f s i is made of G -homogeneous polynomials as well and h f , . . . , f s i is stable by the action of G , see [23, Theo-rem 4]. 14 .3.2. Gr¨obner bases change of orderings We shall say that a zero-dimensional ideal I ⊂ K [ x ] has Property S, if its reduced Gr¨obner basis for lex ( x n ≺ · · · ≺ x ) is in shape position . That is,there exist g , . . . , g n ∈ K [ x n ] of degree at most D − n x Dn + g n ( x n ) , x n − + g n − ( x n ) , . . . , x + g ( x n ) o . Property M, if its reduced Gr¨obner basis for drl ( x n ≺ · · · ≺ x ) satisfies the following condi-tion. For every monomial m in the staircase associated to this Gr¨obner basis, either mx n isin the staircase or it is the leading monomial of some polynomial in this Gr¨obner basis.We assume that a reduced G -homogeneous Gr¨obner basis for drl ( x n ≺ · · · ≺ x n ) satisfyingproperty M is given and the goal is to recover the reduced Gr¨obner basis for lex ( x n ≺ · · · ≺ x n )satisfying property S. By G -homogeneity, the support of each polynomial in the target Gr¨obnerbasis, n x Dn + g n ( x n ) , x n − + g n − ( x n ) , . . . , x + g ( x n ) o , is already known. It is given by the G -degree of its leading monomial, namely x Dn , x n − , . . . , x . Since G is finite, there exists d > x dn has G -degree (0 , . . . ,
0) and there exists δ n , . . . , δ ≥
0, all minimal, suchthat x δ n n has same G -degree as x Dn and x δ i n has same G -degree as x i for 1 ≤ i ≤ n −
1. Therefore,for 1 ≤ i ≤ n , supp g i = ( x δ i n , x δ i + dn , . . . , x δ i + j D − − δ id k dn ) .Thus, the polynomial g n can be computed by solving the following Hankel system x δ nn x δ n + dn ··· x D − dn x δ nn h x δ n n i h x δ n + dn i · · · h x D − d + δ n n i x δ n + dn h x δ n + dn i h x δ n + dn i · · · h x D − d + δ n n i ... ... ... ... x δ n + ⌊ D − − δ nd ⌋ dn = x D − dn h x D − dn i h x D − d + δ n n i · · · h x D − dn i γ + x Dn x δ nn h x D + δ n n i x δ n + dn h x D + δ n + dn i ... ... x D − dn h x D − dn i = . As stated above, table terms h x in i are defined as r T M in with r picked at random. This is doneby computing v = r T , v = v M n , v = v M n , . . . and then extracting the first coordinate of eachvector to simulate the multiplication by .Since, we do not need all the terms but only v δ n , v δ n + d , v δ n + d , . . . , we first compute v δ n and M dn in order to perform big steps. Let us notice that, following [20, 21], by property M, thecolumns of matrix M n are either vectors of the canonical basis or dense vectors. Thus, M dn hasthe same shape as M n with at most max( D , kd ) dense columns, where k is the number of densecolumns in M n . From [23] and the genericity assumption on I , we know we can split M n in | G | matrices of size at most ⌈ D / | G |⌉ . Furthermore, its dense columns are evenly split in the smallmatrices, i.e. the number of dense columns of each small matrix is at most ⌈ k / | G |⌉ . Then, we canmultiply all these small matrices accordingly to obtain the splitting of M dn .15ow, polynomials g , . . . , g n − can be computed by solving a similar Hankel system: x δ in x δ i + dn ··· x δ i + (cid:22) D − − δ id (cid:23) dn x δ nn h x δ i + δ n n i h x δ i + δ n + dn i · · · " x δ i + δ n + j D − − δ id k dn x δ n + dn h x δ i + δ n + dn i h x δ i + δ n + dn i · · · " x δ i + δ n + j D − − δ id + k dn ... ... ... ... x D − dn h x δ i + D − dn i h x δ i + δ n + D − dn i · · · " x δ i + δ n + j D − − δ id − k d + Dn γ + x i x δ nn h x i x δ n n i x δ n + dn h x i x δ n + dn i ... ... x D − dn h x i x D − dn i = . However, the matrices might all be di ff erent. In order to speed up the computation, we changethe linear systems into ones with the same matrix as the first one. This is done by multiplying allthe columns labels by x δ n − δ i n . Proposition 3.7.
Let I ⊂ K [ x ] be a zero-dimensional ideal of degree D, invariant under theaction of a finite diagonal matrix group G. Let us assume that I satisfies both properties S andM and that matrix M n has k dense columns. Let furthermore S be the staircase associated to the lex ( x n ≺ · · · ≺ x ) Gr¨obner basis of I, T ( Λ ≥ ) be the set of monomials of G-degree and forA , B ⊆ T , A + B = { ab | a ∈ A , b ∈ B } be the Minkowski sum of A and B.Then, we can recover the lex ( x n ≺ · · · ≺ x ) Gr¨obner basis, G , of I from its drl ( x n ≺ · · · ≺ x ) Gr¨obner basis using (cid:0) ( S ∩ T ( Λ ≥ )) + (cid:0) ( S ∩ T ( Λ ≥ )) ∪ lm ≺ ( G ) (cid:1)(cid:1) table terms and O (cid:16) kD | G | (cid:17) operations.Proof. Since ideal I satisfies property S, the staircase S associated to its lex ( x n ≺ · · · ≺ x )Gr¨obner basis is n , x n , . . . , x D − n o . Therefore, by definition of d , S ∩T ( Λ ≥ ) = (cid:26) , x dn , . . . , x ⌊ D − d ⌋ dn (cid:27) .Thus, the matrix rows labels are in bijection with a subset of S ∩T ( Λ ≥ ) while the matrix columnslabels and the right-hand side columns labels are in bijection with a subset of ( S ∩ T ( Λ ≥ )) ∪ lm ≺ ( G ). This show that only (cid:0) ( S ∩ T ( Λ ≥ )) + (cid:0) ( S ∩ T ( Λ ≥ )) ∪ lm ≺ ( G ) (cid:1)(cid:1) table terms are re-quired.Since I also satisfies property G , then M n has k dense columns and D − k columns that arevectors of the canonical basis. Furthermore, these dense columns correspond to G -homogeneouspolynomials, so that each of them only has at most O ( D / | G | ) nonzero coe ffi cients. Thus, M n has O ( kD / | G | ) nonzero coe ffi cients. Now, computing v δ n requires 2 δ n multiplications between M n and a vector. Hence v δ n can be computed in O ( δ n kD / | G | ) operations.It remains to compute v δ n + id = r T M δ + idn for all i up to (2 D − d − δ n ) / d by successivemultiplications by M dn . While M dn has max( D , kd ) dense columns, these dense columns stillrepresent G -homogeneous polynomials, thus M dn has O ( kdD / | G | ) nonzero coe ffi cients. Hence,all these vectors can be computed in O ( kD / | G | ) operations.Finally, these linear systems are Hankel of size O ( D / d ) and can be solved in O (cid:16) n M (cid:16) Dd (cid:17) log Dd (cid:17) operations, which is not the bottleneck of the algorithm.Let us note that this extends [23, Theorem 10] where the complexity O ( D ) drops to O (cid:16) D | G | (cid:17) .16 . Adaptive approach A daptive S calar -FGLM algorithm The A daptive S calar -FGLM algorithm aims at computing the minimal Gr¨obner basis ofthe ideal of relations of the input table v for the input monomial ordering ≺ by minimizing thenumber of table queries. To do so, the goal is to build the greatest full-rank multi-Hankel matrixgiven by v increasingly. That is, we start with the empty set S = ∅ . If H S ∪ { x i } , S ∪ { x i } has a greaterrank than H S , S , then S is replaced by S ∪ n x i o . Otherwise we have found a relation with leadingmonomial x i and we shall never try any multiple of x i as a new term in S .In the cone setting, as in Section 3.1, the two strategies can be used. If we build an auxiliarytable w ∈ K N ν , then the A daptive S calar -FGLM algorithm can directly be called on w providedwe only try to add monomials y j that are in T ( N ν ) / I ( C ). If we rather call it on the original table v ∈ K N n , then we modify the algorithm so that only monomials in T ( C ) are used. Furthermore,once a relation with leading monomial x i is found, we shall never try any multiple x i + j in thecone, i.e. with x j ∈ T ( C ). Example 4.1.
Consider the linear King walk v = (cid:0) v i , i (cid:1) ( i , i ) ∈ N counting the number of waysto reach i in i steps of size starting from in the nonnegative ray. It is clear that v i , i = whenever either i > i or i + i = , so that we shall only consider the cone C = n ( i , i ) ∈ N (cid:12)(cid:12)(cid:12) i + i = , i ≤ i o = (1 , N + (0 , N . Assume we consider the lex ( x ≺ x ) ordering, so that T ( C ) = n , x x , x , x x , x , . . . o . We build the matrix (cid:16) (cid:17) which has full rank. We increase the matrix by adding monomials in T ( C ) so we build x x x x ! whichdoes not have full rank, so we have (incorrectly) found that x x − is in the ideal ofrelations. We increase the matrix to x x ! which has full rank. We increase the matrix to x x x x which has full rank. And so on.
In the lattice setting however, we need to be more careful. We shall make one matrix perelement in Z n / Λ and each time we must add an extra column and an extra row, they will beadded to the matrix corresponding to the monomial labeling the extra column. If there is no rankincrease, then as usual a relation is found and no multiple of this monomial will ever label anynew column in any matrix. This yields Algorithm 3 and Theorem 1.3.17 lgorithm 3: L attice A daptive S calar -FGLM Input:
A table v = ( v i ) i ∈ N n with coe ffi cients in K , a monomial ordering ≺ , a nonnegative lattice Λ ⊆ N n , a set A ⊆ N n containing 0 such that Λ + A = Z n . Output:
A set G of relations. If v (0 ,..., = then Return [1]. L ≔ { x , . . . , x n } .Sort L by increasing order wrt. ≺ . G ≔ ∅ // the future set of relations For all a ∈ A do S a ≔ { } . // the future staircase While L , ∅ do m ≔ first element of L and remove it from L .Pick a ∈ A such that m ∈ T (cid:0) ( a + Λ ) ≥ (cid:1) . S ′ ≔ S a ∪ { m } . If H S ′ , S ′ is full rank then // No relation S a ≔ S ′ . L ≔ L ∪ { x m , . . . , x n m } Sort L by increasing order wrt. ≺ and remove duplicates andmultiples of lm ≺ ( G ). Else // Relation!
Solve H S a , S a γ + H S a , { m } = G ≔ G ∪ n m + P s ∈ S a γ s s o and remove multiples of m in L . return G . Proof of Theorem 1.3.
At step m = lm ≺ ( g ), only monomials less than m can have been addedto S a . Thus, the current set S a is actually the final set S a with only elements less than m , i.e. S a ∩ { t ≺ m } . Now, H S a ∩{ t ≺ m } , S a ∩{ t ≺ m } γ + H S a ∩{ t ≺ m } , { m } = (cid:2) gs (cid:3) = s a rowindex, that is s ∈ S a with s ≺ m .Let us prove the second assertion. For any a ∈ A , let S a = S ∩ T (cid:0) ( a + Λ ) ≥ (cid:1) . A necessaryand su ffi cient condition for the L attice A daptive S calar -FGLM algorithm to correctly guess G isthat for each a , S a ⊆ S a , which means that S a = { }∪S a . This can only happen if, for each a andeach monomial m ∈ S a , the rank condition is fulfilled. A su ffi cient condition for this to happenis that matrix H { }∪S a , S a does not have any rank-defect submatrix. This can in turn be translatedinto a nonzero polynomial system, made of the determinants of all the submatrices, whose set ofindeterminates is the matrix coe ffi cients, i.e. the table terms [ s ] with s ∈ S a ∈A { } ∪ S a ) ⊆ S .Hence, there is a non empty Zariski open set of values for these table terms such that S and G are correctly guessed. Remark 4.2.
If an incorrect staircase is guessed, then not much can be said on the output setof polynomials compared to the correct Gr¨obner basis. However, we know that the guessedstaircase is included in the correct one.
Example 4.3.
Let us consider the same table as in Example 3.4, v = (cid:16) i ( j + (cid:17) ( i , j ) ∈ N andits associated lattice Λ = (0 , Z + (1 , Z , so that A = { (0 , , (0 , , (0 , } . We also considerthe lex ( y ≺ x ) ordering. We build three matrices (cid:16) (cid:17) which have full rank. We increase the second matrix to y y ! which has full rank. We increase the third matrix to y y ! which has full rank. We increase the first matrix to y y ! which does not have full rank so that we havefound that y − is in the ideal of relations. We increase the first matrix to x x ! which does not have full rank so that we havefound that x − is in the ideal of relations. We return n y − , x − o .4.2. Mixed approach for guessing P-relations In [6], the authors proposed a mixed approach for guessing P-relations based on a Gr¨obnerbasis computations for reducing the number of table queries. The idea is that if two polynomials g , g ∈ K h t , x i are P-relations satisfied by the table, then any polynomial in h g , g i is also aP-relation. Therefore, as soon as two P-relations g and g are guessed, the goal is to compute aGr¨obner basis { g , g , . . . , g r } of h g , g i . This will yields polynomials, namely g , . . . , g r , whoseleading monomials are not in h lm ≺ ( g ) , lm ≺ ( g ) i . The advantage of this method is two-fold.First, since lm ≺ ( g ) , . . . , lm ≺ ( g r ) ≻ lm ≺ ( g ) , lm ≺ ( g ), they require more queries to the table to becorrectly guessed. Yet, such a Gr¨obner basis computation does not require any more queries.Then, these P-relations may help us determine that the ideal of P-relations is 0-dimensional in K ( t ) h x i . This is a necessary condition for the table to be P-finite.The aim of this section is to extend this approach for guessing P-relations of a table when onlyconsidering terms in a cone or when the ideal of relations is stable by the action of a subgroup ofGL( n ). Lemma 4.4.
Let T ( C ) be a cone of monomials in x , . . . , x n , as before. Let us assume thatf , f ∈ K h t , x i are both polynomials with monomials in T ( N n ) × T ( C ) = n t k x i (cid:12)(cid:12)(cid:12) x i ∈ T ( C ) o . Then,any polynomial f a + f a in the right ideal h f , f i , such that supp a , supp a ∈ T ( N n ) × T ( C ) ,has its support in T ( N n ) × T ( C ) as well.In particular, we can compute a sparse Gr¨obner basis of h f , f i with monomials all in T ( N n ) × T ( C ) using Buchberger’s algorithm or Faug`ere’s F algorithm, restricted to only multi-plying the polynomials by monomials in T ( N n ) × T ( C ) .Proof. We need to prove that if supp f and supp a are in T ( N n ) × T ( C ), then so is supp f a . Bylinearity, this comes down to proving that if two monomials t ℓ x j and t k x i are in T ( N n ) × T ( C ),then so is the support of their product. Since t ℓ x j t k x i = t ℓ ( t − j ) k x j + i = ℓ ,...,ℓ n X q ,..., q n = k q ! · · · k n q n ! ( − j ) k − q · · · ( − j n ) k n − q n t ℓ + q · · · t ℓ n + q n n x j + i x j + i ∈ T ( C ), then t ℓ x i t k x i ∈ T ( N n ) × T ( C ).Now, in T ( N n ) × T ( C ), we can define the division of monomials with m | m if there exists m ∈ T ( N n ) × T ( C ) such that m = m m . Then, we can make a new S-polynomial of twopolynomials with supports in T ( N n ) × T ( C ) by considering the lcm in T ( N n ) × T ( C ) of theirleading monomials.This lemma shows that the definition of sparse Gr¨obner bases and the algorithmic techniquesto compute them in [22] can be extended to skew polynomial rings.Using the definitions and notation of Section 3.3.1, we have the following lemma. Lemma 4.5.
Let G be a finite group of diagonal matrices acting on t , . . . , t n , x , . . . , x n , then Glets t , . . . , t n , each, invariant.Assume that f , f ∈ K h t , x i are both G-homogeneous polynomials, then their S-polynomialis also G-homogeneous. Thus, so are all the elements of a reduced Gr¨obner basis of h f , f i .Proof. There exists a root of unity ζ such that for each matrix in G , there exist integers τ , . . . , τ n ,ε , . . . , ε n such that for all 1 ≤ p ≤ n , x p is sent onto ζ ε p x p and t p onto ζ τ p t p .Therefore, t p x p − x p t p = x p is sent on both ζ τ p t p ζ ε p x p − ζ ε p x p ζ τ p t p = ζ τ p + ε p (cid:16) t p x p − x p t p (cid:17) = ζ τ p + ε p x p and ζ ε p x p . Thus, ζ τ p = G lets t p invariant. By Definition 3.6, this means that the G -degree of t p is 0 so that the t k x i and x i have same G -degree.The S-polynomial of f and f is f t k x i − f lc ≺ ( f ) lc ≺ ( f ) t ℓ x j with t k lm ≺ ( f ) x i = t ℓ lm ≺ ( f ) x j = gcd ( lm ≺ ( f ) , lm ≺ ( f )), where lc ≺ ( f ) stands for leading coe ffi cient of f , i.e. the coe ffi cient of lm ≺ ( f ). Since both terms of the sum have the same leading monomial, it remains to showthat multiplying a polynomial by a monomial preserves the G -homogeneity. Since t ℓ x j t k x i = t ℓ ( t − j ) k x j + i , then it is a G -homogeneous polynomial of same G -degree as x j + i . Now, the G -degree of x j + i is the sum of the G -degrees of x j and x i and thus of t ℓ x j and t k x i .From Lemmas 4.4 and 4.5, we can compute a Gr¨obner basis or a sparse Gr¨obner basis of theideal spanned by skew polynomials associated to P-relations to guess new P-relations in the coneand lattice settings. Corollary 4.6.
Let G be a finite diagonal matrix group acting on variables t and x . Let I = h f , . . . , f s i ⊂ K h t , x i be an ideal spanned by G-homogeneous polynomials. Then, one cancompute a Gr¨obner basis of I by using a quasi-commutative variant of the F algorithm [17]building | G | Macaulay matrices for each G-degree.
5. Experiments
In this section, we report on our implementations of these methods. In Table 1, we considerthe FGLM application running on an I ntel X eon E-2286M with 32 GB of RAM. We computefirst a Gr¨obner basis of an ideal invariant by the action of a finite diagonal group Z / n Z . and thenthe eliminating polynomial of the last variable. The number n in the names of the systems denotesthe number of variables and the computations were done modulo 2 < p < such that a primi-tive n th root of unity exists in Z / p Z . We implemented in C the S parse -FGLM algorithm [20, 21],it generates a scalar table first and then guesses its C-relation with the Berlekamp–Massey algo-rithm. The first part is the bottleneck of the method. In column S parse -FGLM, we use the wholemultiplication matrix, while in column lattice S parse -FGLM, we use the n nonzero blocks of themultiplication matrix to perform the computations and taking advantage of the action of Z / n Z .20e also compare with M aple Groebner:-FGLM to compute a Gr¨obner basisfor an ordering eliminating all the variables but the last one. As expected by Proposition 3.7,using the splitting of the multiplication matrix allows us to divide the computation time by n .Type Degree S parse -FGLM lattice S parse -FGLM M aple Seq. gen. Guess. Seq. gen. Guess.Cyclic-6 156 1 380 70 180 10 120 000Cyclic-7 924 63 000 870 4 550 50 13 s Random-3 294 2 800 260 950 140 510 000Random-4 896 66 000 870 8 100 450 2 000 s Random-6 1656 340 000 1 600 24 000 590 1 200 s Random-5 2000 410 000 2 300 33 000 400 49 s Table 1: FGLM application with the action of Z / n Z (in µ s ). We also implemented the S calar -FGLM algorithm for guessing P-relations of tables inM aple g with steps in { (1 , , (1 , , ( − , , ( − , − } and the 3D-space Walk-43 w of [9] with steps in { ( − , − , − , ( − , − , , ( − , , , (1 , , } .In particular, we restrict ourselves to a subsequence of each where one index is 0. Walks comenaturally with a cone structure: for instance whenever n , n ′ + j , then g n , , j =
0. Likewise,whenever n , n ′ + j + k , then w n , , j , k = C Full Orthant N n Matrix size Queries Relations Matrix size Queries RelationsFake Correct Fake Correct g n , , j ×
441 866 11 0 496 ×
495 946 48 0 g n , , j ×
564 1 174 0 0 1 326 ×
661 1 942 84 0 g n , , j ×
711 1 408 15 8 726 ×
715 1 386 67 0 g n , , j × × w n , , i , j ×
211 430 7 1 220 ×
210 395 24 0 w n , , i , j ×
253 552 2 1 680 ×
267 912 37 0 w n , , i , j ×
400 799 11 6 406 ×
400 771 27 0 w n , , i , j ×
522 1 320 2 6 1 540 ×
589 2 073 68 0
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